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New Charmonium States in QCD Sum Rules: a Concise Review
Marina Nielsena, Fernando S. Navarrab, Su Houng Leec
aInstituto de Fısica, Universidade de Sao Paulo, C.P. 66318, 05389-970 Sao Paulo, SP, BrazilbInstituto de Fısica, Universidade de Sao Paulo, C.P. 66318, 05389-970 Sao Paulo, SP, Brazil
cInstitute of Physics and Applied Physics, Yonsei University, Seoul 120-749, Korea
Abstract
In the past years there has been a revival of hadron spectroscopy. Many interesting new hadron states were discovered experimen-tally, some of which do not fit easily into the quark model. This situation motivated a vigorous theoretical activity. This is a rapidlyevolving field with enormous amount of new experimental information. In the present report we include and discuss data whichwere released very recently. The present review is the first one written from the perspective of QCD sum rules (QCDSR), wherewe present the main steps of concrete calculations and compare the results with other approaches and with experimental data.
∗Corresponding authorEmail addresses:[email protected] (Marina Nielsen),
[email protected] (Fernando S. Navarra),[email protected] (Su Houng Lee)
Preprint submitted to Physics Reports October 27, 2018
Contents
1 Introduction 21.1 New experiments . . . . . . . . . . . . . . . . 21.2 New states . . . . . . . . . . . . . . . . . . . . 31.3 New quark configurations . . . . . . . . . . . . 31.4 New mechanisms of decay and production . . . 3
2 QCD sum rules 42.1 Correlation functions . . . . . . . . . . . . . . 42.2 The OPE side . . . . . . . . . . . . . . . . . . 42.3 The phenomenological side . . . . . . . . . . . 52.4 Choice of currents . . . . . . . . . . . . . . . . 52.5 The mass sum rule . . . . . . . . . . . . . . . 72.6 Numerical inputs . . . . . . . . . . . . . . . . 7
3 The X(3872)meson 83.1 Experiment versus theory . . . . . . . . . . . . 83.2 QCDSR studies ofX(3872) . . . . . . . . . . . 11
3.2.1 Two-point function . . . . . . . . . . . 113.2.2 Three-point function . . . . . . . . . . 14
3.3 Summary forX(3872) . . . . . . . . . . . . . . 163.4 Predictions forXb, Xs, Xs
b . . . . . . . . . . . . 16
4 The Y(JPC = 1−−) family 174.1 Experiment versus theory . . . . . . . . . . . . 174.2 QCDSR studies for theY(JPC = 1−−) states . . 194.3 Summary forY(JPC = 1−−) states . . . . . . . 20
5 The Z+(4430)meson 205.1 Experiment versus theory . . . . . . . . . . . . 205.2 QCDSR calculations forZ+(4430) . . . . . . . 215.3 Summary forZ+(4430) . . . . . . . . . . . . . 215.4 Sum rule predictions forB∗B1 and D∗sD1
molecules . . . . . . . . . . . . . . . . . . . . 21
6 The Z+1 (4050)and Z+2 (4250)states 226.1 Experiment versus theory . . . . . . . . . . . . 226.2 QCDSR calculations . . . . . . . . . . . . . . 22
7 The Y(3930)and Y(4140)states 237.1 Experiment versus theory . . . . . . . . . . . . 237.2 QCDSR calculation forY(3930) andY(4140) . 24
8 The X(3915)and X(4350)states 258.1 QCDSR calculation forX(4350) . . . . . . . . 26
9 The X(3940),Z(3930),X(4160)and Y(4008)states 26
10 Other multiquark states 2610.1 ADsD∗ molecular state . . . . . . . . . . . . . 2610.2 A [cc][ ud] state . . . . . . . . . . . . . . . . . 27
11 Summary 28
12 Appendix: Fierz Transformation 29
1. Introduction
We are approaching the end of a decade which will beremembered as the “decade of the revival of hadron spec-troscopy”. During these years severale+e− colliders started tooperate and produce a large body of experimental information.At the same time new data came from the existingp− p collid-ers and also from thee− p accelerators. In Table 1 we give a listof the new charmonium states observed in these accelerators.
Table 1: Charmonium states observed in the last years.
state production mode decay mode ref.X(3872) B→ KX(3872) J/ψππ [1]X(3915) γγ → X(3915) J/ψω [2]Z(3930) γγ → Z(3930) DD [3]Y(3930) B→ KY(3930) J/ψω [4]X(3940) e+e− → J/ψX(3940) DD∗ [5]Y(4008) e+e− → γIS RY(4008) J/ψππ [6]Z+1 (4050) B0→ K−Z+1 (4050) χc1π
+ [7]Y(4140) B→ KY(4140) J/ψφ [8]X(4160) e+e− → J/ψX(4160) D∗D∗ [9]Z+2 (4250) B0→ K−Z+1 (4250) χc1π
+ [7]Y(4260) e+e− → γIS RY(4260) J/ψππ [10]X(4350) γγ → X(4350) J/ψφ [11]Y(4360) e+e− → γIS RY(4260) ψ′ππ [12]Z+(4430) B0→ K−Z+(4430) ψ′π+ [13]X(4630) e+e− → γIS RX(4630) Λ+Λ− [14]Y(4660) e+e− → γIS RY(4660) ψ′ππ [12]
In what follows we will review and comment all this infor-mation [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35,36, 37, 38, 39, 40, 41].
The study of spectroscopy and the decay properties of theheavy flavor mesonic states provides us with useful informa-tion about the dynamics of quarks and gluons at the hadronicscale. The remarkable progress on the experimental side, withvarious high energy machines has opened up new challenges inthe theoretical understanding of heavy flavor hadrons.
1.1. New experiments
The B-factories, the PEPII at SLAC in the U.S.A., and theKEKB at KEK in Japan, were constructed to test the StandardModel mechanism for CP violation. However, their most in-teresting achievement was to contribute to the field of hadronspectroscopy, in particular in the area of charmonium spec-troscopy. They aree+e− colliders operating at a CM energynear 10,580 MeV. TheBB pairs produced are measured by theBaBar (SLAC) and Belle (KEK) collaborations. Charmoniumstates are copiously produced at theB-factories in a variety ofprocesses. At the quark level, theb quark decays weakly to ac quark accompanied by the emission of a virtualW− boson.Approximately half of the time, theW− boson materializes as asc pair. Therefore, half of theB meson decays result in a finalstate that contains acc pair. When thesecc pairs are produced
2
close to each other in phase space, they can coalesce to form acc charmonium meson.
Figure 1: Illustration for the initial state radiation (ISR) process.
The simplest charmonium producingB meson decay is:B→K(cc). Another interesting form to produce charmonium inB-factories is directly from thee+e− collision, when the initialstatee+ or e− occasionally radiates a high energyγ-ray, and thee+e− subsequently annihilate at a corresponding reduced CMenergy, as ilustrated in Fig. 1.
When the energy of the radiatedγ-ray (γIS R) is between 4000and 5000 MeV, thee+e− annhilation occurs at CM energies thatcorrespond to the range of mass of the charmonium mesons.Thus, the initial state radiation (ISR) process can directly pro-duce charmonium states withJPC = 1−−.
1.2. New states
Many states observed by BaBar and Belle collaborations, likethe X(3872),Y(3930),Z(3930),X(3940),Y(4008),Z+1 (4050),Y(4140),X(4160),Z+2 (4250),Y(4260),Y(4360),Z+(4430) andY(4660), remain controversial. A common feature of thesestates is that they are seen to decay to final states that containcharmed and anticharmed quarks. Since their masses and decaymodes are not in agreement with the predictions from poten-tial models, they are considered as candidates for exotic states.By exotic we mean a more complex structure than the sim-ple quark-antiquark state, like hybrid, molecular or tetraquarkstates. The idea of unconventional quark structures is quite oldand despite decades of progress, no exotic meson has been con-clusively identified. In particular, those withqq quantum num-bers should mix with ordinary mesons and are thus hard to un-derstand. Therefore, the observation of these new states isachallenge to our understanding of QCD.
In 2003 a great deal of attention was given to pentaquarkstates, which may be exotic objects [42, 43]. After many nega-tive results the study of these states was abandoned. However,in that same year the discovery by BELLE of the unusual char-monium state named X prompted a lively and deep discussionabout the existence of exotic states in the charmonium sector(i.e. non purec − c states), exactly where, up to that moment,the calculations based on potential models had worked so well.
Establishing the existence of these states means already a re-markable progress in hadron physics. Moreover it poses thenew question: what is the structure of these new states? Thedebate on exotic hadron structure was strongly revived during
the short “pentaquark era”. Unfortunately, because of the un-certainties on the experimental side, it was very soon aborted.However, some interesting ideas were passed along to the sub-sequent discussion about the nature of theX, Y andZ states,which is still in progress.
1.3. New quark configurationsConcerning the structure we can say that there are still at-
tempts to interprete the new states asc− c. One can pursue thisapproach introducing corrections in the potential, such asquarkpair creation. This “screened potential” changes the previousresults, obtained with the unscreened potential and allowstounderstand some of the new data in thec− c approach [44]. De-parting from thec − c assignment, the next option is a systemcomposed by four quarks, which can be uncorrelated, form-ing a kind of bag, or can be grouped in diquarks which thenform a bound system [45, 46, 47]. These configurations arecalled tetraquarks. Alternatively, these four quarks can formtwo mesons which then interact and form a bound state. If themesons contain only one charm quark or antiquark, this con-figuration is referred to as a molecule [48, 49, 50]. If one ofthe mesons is a charmonium, then the configuration is calledhadro-charmonium [51]. Another possible configuration is ahybrid charmonium [52, 53]. In this case, apart from thec− cpair, the state contains excitations of the gluon field. In someimplementations of the hybrid, the excited gluon field is repre-sented by a “string” or flux tube, which can oscillate in normalmodes [50].
The configurations mentioned in the previous paragraph arequite different and are governed by different dynamics. Inquarkonia states the quarks have a short range interaction domi-nated by one gluon exchange and a long range non-perturbativeconfining interaction, which is often parametrized by a linearattractive potential. In tetraquarks besides these two types of in-terations, we may have a diquark-antidiquark interaction,whichis not very well known. In molecules and hadro-charmoniumthe interaction occurs through meson exchange. Finally, insome models inspired by lattice QCD results, there is a fluxtube formation between color charges and also string junctions.With these building blocks one can construct very complicated“stringy” combinations of quarks and antiquarks and their in-teractions follow the rules of string fusion and/or recombination[54, 55]. In principle, the knowledge of the interaction shouldbe enough to determine the spatial configuration of the system.In practice, this is only feasible in simple cases, such as thecharmonium in the non-relativistic approach, where havingthepotential one can solve the Schrodinger equation and determinethe wave function. In other approaches the spatial configurationmust be guessed and it may play a crucial role in the productionand decay of these states.
1.4. New mechanisms of decay and productionThe theoretical study of production and decays of the new
quarkonium states is still in the beginning. Some of the statesare quite narrow and this is difficult to understand in some ofthe theoretical treatments, as for example, in the molecular ap-proach.
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The narrowness of a state might be, among other possibili-ties, a consequence of its spatial configuration. Assuming,forexample, that theX(3872) is a set ofcc uu quarks confined ina bag but with sufficient (or nearly sufficient) energy to decayinto the two mesonsD andD
∗, why is it so difficult for them to
do so? What mechanism hinders this seemingly simple coales-cence process, which is observed in other hadronic reactions?In a not so distant past the same questions was asked in thecase of theΘ+ pentaquark. Now we know that the existenceof this particle is, to say the least, not very likely. Neverthe-less, as mentioned in a previous subsection, some of the ideasadvanced in the pentaquark years might be retaken now in adifferent context to answer the question raised above. In fact,the pentaquark decayΘ+(uudds) → n(udd) + K+(us) was en-ergetically “superallowed”, since there was enough phase spaceavailable and no process of quark pair creation nor annihilation.Why was this quark rearrangement and hadronization difficult?It has been conjectured [56] that theΘ+ was in a diquark (ud)- diquark (ud) - antidiquark (s) configuration, such that, due tothe not very well understood repulsive and/or attractive diquarkinteractions, the two diquarks were far apart from each otherwith the antiquark standing in the middle. In this configurationit was difficult for one diquark to capture the missingd quark (toform a neutron), which was very far. These words were imple-mented in a quantitative model and this configuration was bap-tized “peanut”, since the three bodies were nearly aligned.An-other pentaquark spatial configuration with small decay widthwas the equilateral tetrahedron with four quarks located inthecorners and the antiquark in the center [57]. An other interest-ing conjecture based on lattice arguments was that, due to thedynamics of string rearrangement, the pentaquark constructedwith three string junctions had, during the decay process, topass through a very energetic configuration. Since its initialenergy would be much less than that, this passage would haveto proceed through tunneling and would therefore be stronglysuppressed [54].
These ideas are relevant for the new charmonium physics be-cause if they are multiquark states, as it seems to be the case,their spatial configuration will play a more important role bothin production and decay.
The production of some of these states (such as, for exam-ple, theX) has been observed both ine+ e− and p p colliders,the latter being much more energetic. TheX produced in elec-tron - positron collisions comes fromB decays, whereas inp preactions it must come from a hard gluon splitting into ac cpair, which then undergoes some complicated fragmentationprocess. In theoretical simulations [58], it was shown thatitis very difficult to produceX if it is composed by two boundmesons.
Some reviews about these states can be found in the literature[60, 61, 62, 63, 64, 65, 66, 67, 68]. There are at least two rea-sons for writting a report on the subject. The first one is becausethis is a rapidly evolving field with enormous amount of newexperimental information coming from the analysis of BELLEand BABAR accumulated data and also from CLEO and BESwhich are still running and producing new data. New data arealso expected to come from the LHCb, which will start to op-
erate soon and will be generating data with very high statisticsfor the next several years. In the present report we include anddiscuss data which were not yet available to the previous re-viewers.
The other reason which motivates us to write this report isthat, on the theory side each review is biased and naturally em-phasizes the approach followed by the authors. Thus, in Refs.[60] and [66] the authors present the available data and thendiscuss their theoretical interpretations making a sketchof theexisting theories. In [60] attention is given to the conventionalquark antiquark potential model and to models of the interac-tion between mesons which form a molecular state. In [66]a very nice overview of different theoretical approaches suchas potential models, QCD sum rules and lattice QCD was pre-sented. The author works out some pedagogical examples, us-ing general considerations and simplified assumptions. Thepresent review is the first one written from the perspective ofQCD sum rules, where we present the main steps of concretecalculations and compare the results with other approachesandwith experimental data.
In the next sections we discuss the experimental data and thepossible interpretations for the recently observedX, Y andZmesons.
2. QCD sum rules
2.1. Correlation functions
QCD sum rules are discussed in many reviews [69, 70, 71,72] emphasizing various aspects of the method. The basic ideaof the QCD sum rule formalism is to approach the bound stateproblem in QCD from the asymptotic freedom side,i.e., tostart at short distances, where the quark-gluon dynamics ises-sentially perturbative, and move to larger distances wherethehadronic states are formed, including non-perturbative effects“step by step”, and using some approximate procedure to ex-tract information on hadronic properties.
The QCD sum rule calculations are based on the correlator oftwo hadronic currents. A generic two-point correlation functionis given by
Π(q) ≡ i∫
d4x eiq·x〈0|T[ j(x) j†(0)]|0〉 , (1)
where j(x) is a current with the quantum numbers of the hadronwe want to study.
The fundamental hypothesis of the sum rules approach is theassumption that there is an interval inq over which the corre-lation function may be equivalently described at both the quarkand the hadron levels. The former is called QCD or OPE sideand the latter is called phenomenological side. By matchingthe two sides of the sum rule we obtain information on hadronproperties.
2.2. The OPE side
At the quark level the complex structure of the QCD vac-uum leads us to employ the Wilson’s operator product expan-sion (OPE) [73].
4
In QCD we only know how to work analytically in the per-turbative regime. Therefore, the perturbative part ofΠ(q) inEq.(1) can be reliably calculated. However, this does not yetimply that all important contributions to the QCD side of thesum rule have been taken into account. The complete calcula-tion has to include the effects due to the fields of soft gluons andquarks populating the QCD vacuum. A practical way to calcu-late the vacuum-field contributions to the correlation function isthrough a generalized Wilson OPE. To apply this method to thecorrelation function (1), one has to expand the product of twocurrents in a series of local operators:
Π(q) = i∫
d4x eiq·x〈0|T[ j(x) j†(0)|0〉 =∑
n
Cn(Q2)On , (2)
where the set{On} includes all local gauge invariant operatorsexpressible in terms of the gluon fields and the fields of lightquarks. Eq. (2) is a concise form of the Wilson OPE. The co-efficientsCn(Q2) (Q2 = −q2), by construction, include only theshort-distance domain and can, therefore, be evaluated pertur-batively. Non-perturbative long-distance effects are containedonly in the local operators. In this expasion, the operatorsareordered according to their dimensionn. The lowest-dimensionoperator withn = 0 is the unit operator associated with the per-turbative contribution:C0(Q2) = Πper(Q2), O0 = 1. The QCDvacuum fields are represented in (2) in the form of vacuum con-densates. The lowest dimension condensates are the quark con-densate of dimension three:O3 = 〈qq〉, and the gluon conden-sate of dimension four:O4 = 〈g2G2〉. For non exotic mesons,i.e. normal quark-antiquark states, the contributions of conden-sates with dimension higher than four are suppressed by largepowers ofΛ2
QCD/Q2, where 1/ΛQCD is the typical long-distance
scale. Therefore, even at intermediate values ofQ2 (∼ 1 GeV2),the expansion in Eq. (2) can be safely truncated after dimensionfour condensates. However, for molecular or tetraquark states,the mixed-condensate of dimension five:O5 = 〈qgσ.Gq〉, thefour-quark condensate of dimension six:O6 = 〈qqqq〉 and eventhe quark condensate times the mixed-condensate of dimensioneight: O8 = 〈qqqgσ.Gq〉, can play an important role. The three-gluon condensate of dimension-six:O6 = 〈g3G3〉 can be safelyneglected, since it is suppressed by the loop factor 1/16π2.
In the case of the (dimension six) four-quark condensateand the (dimension eight) quark condensate times the mixed-condensate, in general factorization assumption is assumed andtheir vacuum saturation values are given by:
〈qqqq〉 = 〈qq〉2, 〈qqqgσ.Gq〉 = 〈qq〉〈qgσ.Gq〉. (3)
Their precise evaluation requires more involved analysis in-cluding a non-trivial choice of factorization scheme [74].Inorder to account for deviations of the factorization hypothesis,we will use the parametrization:
〈qqqq〉 = ρ〈qq〉2, 〈qqqgσ.Gq〉 = ρ〈qq〉〈qgσ.Gq〉, (4)
whereρ = 1 gives the vacuum saturation values andρ = 2.1indicates the violation of the factorization assumption [71, 75,76].
2.3. The phenomenological side
The generic correlation function in Eq. (1) has a dispersionrepresentation
Π(q2) = −∫
dsρ(s)
q2 − s+ iǫ+ · · · , (5)
through its discontinuity,ρ(s), on the physical cut. The dots inEq. (5) represent subtraction terms.
The discontinuity can be written as the imaginary part of thecorrelation function:
ρ(s) =1π
Im[Π(s)] . (6)
The evaluation of the spectral density (ρ(s)) is simpler than theevaluation of the correlation function itself, and the knownledgeof ρ(s) allows one to recover the whole functionΠ(q2) throughthe integral in Eq. (5).
The calculation of the phenomenological side proceeds byinserting intermediate states for the hadron,H, of interest.The currentj ( j†) is an operator that annihilates (creates) allhadronic states that have the same quantum numbers asj. Con-sequently,Π(q) contains information about all these hadronicstates, including the low mass hadron of interest. In order forthe QCD sum rule technique to be useful, one must parametrizeρ(s) with a small number of parameters. The lowest resonanceis often fairly narrow, whereas higher-mass states are broader.Therefore, one can parameterize the spectral density as a singlesharp pole representing the lowest resonance of massm, plus asmooth continuum representing higher mass states:
ρ(s) = λ2δ(s−m2) + ρcont(s) , (7)
whereλ gives the coupling of the current with the low masshadron,H:
〈0| j|H〉 = λ. (8)
For simplicity, one often assumes that the continuum contri-bution to the spectral density,ρcont(s) in Eq. (7), vanishes bel-low a certain continuum thresholds0. Above this threshold, itis assumed to be given by the result obtained with the OPE.Therefore, one uses the ansatz
ρcont(s) = ρOPE(s)Θ(s− s0) . (9)
2.4. Choice of currents
Mesonic currents for open charm mesons are given in Ta-ble 2.
Table 2: Currents for theD mesonsstate symbol current JP
scalar meson D0 qc 0+
pseudoscalar meson D iqγ5c 0−
vector meson D∗ qγµc 1−
axial-vector meson D1 qγµγ5c 1+
From these currents we can construct molecular currentswhich can be eigenstates of charge conjugationC andG-parity.
5
Let us consider, as an example, a current withJPC = 1++ for themolecularD0D∗0 system. It can be written as a combination oftwo currents [77, 78]:
j1µ(x) = [u(x)γ5c(x)][ c(x)γµu(x)], (10)
andj2µ(x) = [u(x)γµc(x)][ c(x)γ5u(x)]. (11)
Since the charge conjugation transformation is defined as:(q)C = −qTC−1 = qTC and (q)C = CqT , we get
( j1µ)C = −(cγ5u)(uγµc) = − j2µ, (12)
( j2µ)C = −(cγµu)(uγ5c) = − j1µ. (13)
Therefore, the current
jµ(x) =1√
2
(
j1µ(x) − j2µ(x))
, (14)
has positiveC. However, this current is not aG-parity eigen-state. TheG-parity transformation is an isospin rotation of thecharge conjugated current:
( j1µ)G = −(cγ5d)(dγµc), (15)
( j2µ)G = −(cγµd)(dγ5c). (16)
In the case of a charged molecularD1D∗ current withJP =
0−, it can also be written as a combination of two currents:
j1 = (cγµγ5u)(dγµc), (17)
j2 = (cγµu)(dγµγ5c). (18)
The charge conjugation transformation in these currents leadsto
( j1)C = −(uγµγ5c)(cγµd), (19)
( j2)C = −(uγµc)(cγµγ5d), (20)
and the isospin rotation gives
( j1)G = (dγµγ5c)(cγµu) = j2, (21)
( j2)G = (dγµc)(cγµγ5u) = j1. (22)
Therefore, the current
j =1√
2( j1 + j2) , (23)
has positiveG-parity.In the case of tetraquark [cq][ cq] currents, they can be
constructed in terms of color anti-symmetric diquark states:εabc[qT
aCΓcb], where a, b, c are color indices of theS U(3)color group,C is the charge conjugation matrix, andΓ standsfor Dirac matrices. The quantum numbers of the diquark statesare given in Table 3.
From these diquarks we can construct tetraquark currentswhich can be eigenstates of charge conjugationC andG-parity.
Table 3: Currents for charmed diquark states.
state current JP
scalar diquark qTaCγ5cb 0+
pseudoscalar diquark qTaCcb 0−
vector diquark qTaCγ5γµcb 1−
axial-vector diquark qTaCγµcb 1+
In the case of aJPC = 1++ current it can be written as a combi-nation of scalar and axial-vector diquarks:
j1µ = ǫabcǫdec(qTaCγ5cb)(qdγµCcT
e ), (24)
andj2µ = ǫabcǫdec(qT
aCγµcb)(qdγ5CcTe ). (25)
It is interesting to notice that the structure of the currents inEqs. (24) and (25) relates the spin of the charm quark with thespin of the light quark. This is very different from the spin struc-ture of the heavy quark effective theory [79]. Heavy quark ef-fective theory, in leading order in 1/M, possesses a heavy-quarkspin symmetry. Therefore, hadrons can be classified accord-ing to the angular momentum and parity of the light fields only.This gives 0− and 1− mesons with identical properties.
Using the charge conjugation transformations one gets:
( j1µ)C = ǫabcǫdec(qaγ5CcTb )(qT
dCγµce) = j2µ, (26)
( j2µ)C = ǫabcǫdec(qaγµCcTb )(qT
dCγ5ce) = j1µ. (27)
Therefore, the current
jµ(x) =i√
2
(
j1µ(x) + j2µ(x))
, (28)
has positiveC. Thei was used in Eq. (28) to insure thatj†µ = jµ.As in the case of theD0D∗0 molecular current, the current inEq. (28) is not aG-parity eigenstate. However, other combina-tions of tetraquark currents can be constructed to beG-parityeigenstates.
In general, there is no one to one correspondence between thecurrent and the state, since the current in Eq. (28) can be rewrit-ten in terms of a sum over molecular type currents through theFierz transformation. In the appendix, we provide the generalexpressions for the Fierz transformation of the tetraquarkcur-rents into molecular type of currents as given in Eq. (14). How-ever, as shown in the appendix, in the Fierz transformation of atetraquark current, each molecular component contributeswithsuppression factors that originate from picking up the correctDirac and color indices. This means that if the physical state isa molecular state, it would be best to choose a molecular typeof current so that it has a large overlap with the physical state.Similarly for a tetraquark state it would be best to choose atetraquark current. If the current is found to have a large over-lap with the physical state, the range of Borel parameters wherethe pole dominates over the continuum would be larger, andthe OPE for the mass would have a better convergence. Theseconditions will lead to a better sum rule; this means that the
6
Borel curve has an extremum or is flat, and the calculated massis close to the physical value. Therefore, if the sum rule givesa mass and width consistent with the physical values, we caninfer that the physical state has a structure well represented bythe chosen current. In this way, we can indirectly discriminatebetween the tetraquark and the molecular structures of the re-cently observed states. However, it is very important to noticethat since the molecular currents, as the one in Eq. (14), arelo-cal, they do not represent extended objects, with two mesonsseparated in space, but rather a very compact object with twosinglet quark-antiquark pairs.
When the current is fixed, we proceed by inserting it intoEq. (1). Contracting all the quark anti-quark pairs, we canrewrite the correlation function in terms of the quark propa-gators, and then we can perform the OPE expansion of thesepropagators. For the light quarks, keeping terms which are lin-ear in the light quark massmq, this expansion reads:
Sab(x) = 〈0T[qa(x)qb(0)]〉0 = iδab
2π2x4x/ −
mqδab
4π2x2
−tAabgGA
µν
32π2
( ix2
(x/σµν + σµνx/) −mqσµν ln(−x2)
)
− δab
12〈qq〉 + iδab
48mq〈qq〉x/ − x2δab
26 × 3〈qgσ.Gq〉
+ix2δab
27 × 32mq〈qgσ.Gq〉x/, (29)
where we have used the fixed-point gauge. For heavy quarks,it is more convenient to work in the momentum space. In thiscase the expansion is given by:
Sab(p) = ip/ +m
p2 −m2δab−
i4
tAabgGAµν[σ
µν(p/ +m) + (p/ +m)σµν]
(p2 −m2)2
+iδab
12m〈g2G2〉 p2 +mp/
(p2 −m2)4(30)
2.5. The mass sum ruleNow one might attempt to match the two descriptions of the
correlator:Πphen(Q2)↔ ΠOPE(Q2) . (31)
However, such a matching is not yet practical. The OPE sideis only valid at sufficiently large spacelikeQ2. On the otherhand, the phenomenological description is significantly domi-nated by the lowest pole only for sufficiently smallQ2, or betteryet, timelikeq2 near the pole. To improve the overlap betweenthe two sides of the sum rule, one applies the Borel transforma-tion
BM2[Π(q2)] = lim−q2,n→∞−q2/n=M2
(−q2)n+1
n!
(
ddq2
)n
Π(q2) . (32)
Two important examples are:
BM2
[
q2n]= 0 , (33)
and
BM2
[
1(m2 − q2)n
]
=1
(n− 1)!e−m2/M2
(M2)n−1, (34)
for n > 0. From these two results, (33) and (34), one can seethat the Borel transformation removes the subtraction terms inthe dispersion relation, and exponentially suppresses thecontri-bution from excited resonances and continuum states in the phe-nomenological side. In the OPE side the Borel transformationsuppresses the contribution from higher dimension condensatesby a factorial term.
After making a Borel transform on both sides of the sum rule,and transferring the continuum contribution to the OPE side, thesum rule can be written as
λ2e−m2/M2=
∫ s0
smin
ds e−s/M2ρOPE(s) . (35)
If both sides of the sum rule were calculated to arbitrary highaccuracy, the matching would be independent ofM2. In prac-tice, however, both sides are represented imperfectly. Thehopeis that there exists a range ofM2, called Borel window, in whichthe two sides have a good overlap and information on the low-est resonance can be extracted. In general, to determine theallowed Borel window, one analyses the OPE convergence andthe pole contribution: the minimum value of the Borel mass isfixed by considering the convergence of the OPE, and the max-imum value of the Borel mass is determined by imposing thecondition that the pole contribution should be bigger than thecontinuum contribution.
In order to extract the massm without worrying about thevalue of the couplingλ, it is possible to take the derivative ofEq. (64) with respect to 1/M2, and divide the result by Eq. (64).This gives:
m2 =
∫ s0
sminds e−s/M2
sρOPE(s)∫ s0
sminds e−s/M2
ρOPE(s). (36)
This quantity has the advantage to be less sensitive to the per-turbative radiative corrections than the individual sum rules.Therefore, we expect that our results obtained to leading orderin αs will be quite accurate.
2.6. Numerical inputs
In the following sections we will present numerical results.In the quantitative aspect, QCDSR is not like a model whichcontains free parameters to be adjusted by fitting data. The in-puts for numerical evaluations are the following:i) the vac-uum matrix elements of composite operators involving quarksand gluons which appear in the operator product expansion.These numbers, known as condensates, contain all the non-perturbative component of the approach. They could in prin-ciple, be calculated in lattice QCD. In practice they are esti-mated phenomenologically. They are universal and, once ad-justed to fit, for example, the mass of a particle, they must havealways that same value. They are the analogue for spectroscopyof the parton distribution functions in deep inelastic scatter-ing; ii ) quark masses, which are extracted from many differ-ent phenomenological analyses and used in our calculation;iii )the threshold parametrs0 is the energy (squared) which charac-terizes the beginning of the continuum. Typically the quantity
7
√s0−m (wherem is the mass of the ground state particle) is the
energy needed to excite the particle to its first excited state withthe same quantum numbers. This number is not well known, butshould lie between 0.3 and 0.8 GeV. If larger deviations fromthis interval are needed, the calculation becomes less reliable.
All in all, in QCDSR we do not have much freedom forchoosing numbers. In the calculations discussed in the nextsections we will use [71, 80, 81, 82]:
ms = (0.13± 0.03) GeV,
mc(mc) = (1.23± 0.05) GeV,
mb(mb) = (4.24± 0.06) GeV,
〈qq〉 = −(0.23± 0.03)3 GeV3,
〈ss〉 = (0.8± 0.2)〈qq〉,〈g2G2〉 = 0.88 GeV4,
〈qgσ.Gq〉 = m20〈qq〉, m2
0 = 0.8 GeV2. (37)
3. The X(3872) meson
Since its first observation in August 2003 by Belle Collab-oration [1], theX(3872) represents a puzzle and, up to now,there is no consensus about its structure. TheX(3872) has beenconfirmed by CDF, D0 and BaBar [15, 16, 17]. Besides the dis-covery production modeB+→ X(3872)K+ → J/ψπ+π−K+, theX(3872) has been observed in pronptpp production [15, 16].However, searches in prompte+e− production [18] and ine+e−
or γγ formation [19] have given negative results so far. Thecurrent world average mass is
MX = (3871.20± 0.39) MeV , (38)
and the most precise measurement to date isMX = (3871.61±0.16± 0.19) MeV, as can be seen by Fig.2 [20].
)2X(3872) Mass ( MeV/c3866 3867 3868 3869 3870 3871 3872 3873
*)o)+m(Dom(D2 0.36 MeV/c±3871.81
new average2 0.22 MeV/c±3871.51
old average2 0.39 MeV/c±3871.20
CDF new (preliminary)2 0.19 MeV/c± 0.16 ±3871.61
CDF old2 0.40 MeV/c± 0.70 ±3871.30
D02 3.00 MeV/c± 3.10 ±3871.80
)0BaBar (B2 0.20 MeV/c± 1.20 ±3868.60
)+BaBar (B2 0.10 MeV/c± 0.60 ±3871.30
Belle2 0.50 MeV/c± 0.60 ±3872.00
X(3872) Mass Measurements
Figure 2: An overview of the measuredX(3872) masses from ref. [20].
The mass of theX(3872) is at the threshold for the productionof the charmed meson pairm(D0D0∗) = 3871.81± 0.36 MeV[21], and this state is extremely narrow: its width is less than2.3 MeV at 90% confidence level.
3.1. Experiment versus theory
Both Belle and Babar collaborations reported the radiativedecay modeX(3872)→ γJ/ψ [22, 23], which determinesC =+. Belle Collaboration reported the branching ratio:
Γ(X→ J/ψγ)Γ(X→ J/ψ π+π−)
= 0.14± 0.05. (39)
Recent studies from Belle and CDF that combine angular in-formation and kinematic properties of theπ+π− pair, stronglyfavor the quantum numbersJPC = 1++ [22, 24, 25]. In particu-lar, in ref. [25] it was shown that only the hypothesesJPC = 1++
and 2−+ are compatible with data. All other possible quantumnumbers are ruled out by more than three standard deviations.The possibility 2−+ is disfavored by the observation of the decayinto ψ(2S)γ [26] and also by the observation of the decays intoD0D0π0 by Belle and BaBar Collaborations [27, 28]. On theother hand, the possibility 1++ is disfavored by the observationof the decay intoJ/ψω by BaBar Collaborations [83]. In thefollowing we will asume the quantum numbers of theX(3872)to be 1++.
From constituent quark models [84] the masses of the possi-ble charmonium states withJPC = 1++ quantum numbers are:2 3P1(3925) and 33P1(4270), and lattice QCD calculations give2 3P1(4010) [85] and 23P1(4067) [86]. In all cases the pre-dictons for the mass of the 23P1 charmonium state are muchbigger than the observed mass. However, a more recent latticeQCD calculation gives 23P1(3853) [87] and, therefore, this in-terpretation can not be totally discarded [88]. In any case,thestrongest fact against thecc assignment forX(3872) [84, 89], isthe fact that from the study of the dipion mass distribution in theX(3872)→ J/ψπ+π− decay, Belle [1] and CDF [24] concludedthat it proceeds through theX(3872)→ J/ψρ0 decay. Since acharmonium state has isospin zero, it can not decay easily intoa J/ψρ final state.
The proximity between theX mass and theD∗0D0 thresh-old inspired the proposal that theX(3872) could be a molecularbound state with small binding energy [48, 49, 50, 90, 91, 92,93]. As a matter of fact, Tornqvist, using a meson potentialmodel [94], essentially predicted theX(3872) in 1994, since hefound that there should be molecules near theD∗D thresholdin the JPC = 0−+ and 1++ channels. The only other molecularstate that is predicted in the potential model updated by Swan-son [60] is a 0++ D∗D∗ molecule at 4013 MeV.
In ref. [50] Swanson proposed that theX(3872) could bemainly a D0D∗0 molecule with a small but important admix-ture ofρJ/ψ andωJ/ψ components. With this picture the decaymodeX(3872)→ J/ψ π+π−π0 was predicted at a rate compa-rable to theX(3872)→ J/ψπ+π− mode. Soon after this pre-diction, Belle Coll. [22] reported the observation of thesetwodecay modes at a rate:
B(X→ J/ψ π+π−π0)B(X→ J/ψπ+π−)
= 1.0± 0.4± 0.3. (40)
This observation establishes strong isospin and G parity viola-tion and strongly favors a molecular assignment forX. How-ever, it still does not completely exclude acc interpretation for
8
X since the isospin and G parity non-conservation in Eq. (40)could be of dynamical origin due toρ0−ω mixing [95] or evendue to final state interactions (FSI) containingD loops, such asX→ J/ψω → DD → J/ψρ. Although all the ingredients (spe-cially the charm form factors [96, 97]) for the relevant effectivefield theory are available, there are no quantitative results in theFSI approach yet.
The decayX→ J/ψω was also observed by BaBar Collabo-ration [83] at a rate:
B(X→ J/ψπ+π−π0)B(X→ J/ψπ+π−)
= 0.8± 0.3, (41)
which is consistent with the result in Eq. (40).It is also important to notice that, although aD0D∗0 molecule
is not an isospin eingenstate, the ratio in Eq. (40) can not bereproduced by a pureD0D∗0 molecule. In ref. [98] it was shownthat for a pureD0D∗0 molecule
Γ(X(D0D∗0)→ J/ψ π+π−π0)
Γ(X(D0D∗0)→ J/ψ π+π−)≃ 0.15. (42)
In refs. [27, 28] Belle and BaBar Collaborations reported theobservation of a near threshold enhancement in theD0D0π0 sys-tem. The peak mass values for the two observations are in goodagreement with each other: (3875.2± 1.9) MeV for Belle and(3875.1± 1.2) MeV for BaBar, and are higher than in the massof theX(3872) observed in theJ/ψπ+π− channel by (3.8± 1.1)MeV. Since this peak lies about 3 MeV above theD∗0D0 thresh-old, it is very awkward to treat it as aD∗0D0 bound state. Ac-cording to Braaten [99], the larger mass of theX measured intheD0D0π0 decay channel could be explained by the differencebetween the line shapes of theX into the two decays:D0D0π0
andJ/ψπ+π−. In the decay of a narrowX molecular state intoits constituentsD∗0D0, the width ofD∗0 distorts the decay lineshape of theX(3872) [100]. Therefore, the peak observed intheB→ K D0D0π0 decay channel could be a combination of aresonance below theD∗0D0 threshold from theB→ K X decayand a threshold enhancement above theD∗0D0 threshold. In thiscase, fitting theD0D0π0 invariant mass to a Breit-Wigner doesnot give reliable values for the mass and width. However, in anew measurement [30] Belle has obtained a mass (3872.6±0.6)MeV in theD0D∗0 invariant mass spectrum, which is consistentwith the current world average mass forX(3872). Using thisnew data and taking into account the universal features of theS-wave threshold resonance Braaten and Stapleton concludedthat theX(3872) is a extremely weakly-bound charm mesonmolecule [101].
Other interesting possible interpration of theX(3872), firstproposed by Maianiet al. [45], is that it could be a tetraquarkstate resulting from the binding of a diquark and an antidiquark[95]. This construction is based in the idea that diquarks canform bound-states, which can be treated as confined particles,and used as degrees of freedom in parallel with quarks then-selves [102, 103, 104]. Therefore, the tetraquark interpreta-tion differs from the molecular interpretation in the way thatthe quarks are organized in the state, as shown in Fig. 1. Thedrawback of the tetraquark picture is the proliferation of the
Figure 3: Cartoon representation for the molecular and tetraquark interpreta-tions ofX(3872).
predicted states [45] and the lack of selection rules that couldexplain why many of these states are not seen [105].
The authors of ref. [45] have considered diquark-antidiquarkstates withJPC = 1++ and symmetric spin distribution:
Xq = [cq]S=1[cq]S=0 + [cq]S=0[cq]S=1. (43)
The most general states that can decay into 2π and 3π are:
Xl = cosθXu + sinθXd, Xh = cosθXd − sinθXu. (44)
Imposing the rate in Eq.(40), they getθ ∼ 200. It is important tonotice that a similar mixture betweenD0D∗0 andD+D∗− molec-ular states with the same mixing angleθ ∼ 200 [98], would alsoreproduce the decay rate in Eq.(40).
The authors of ref. [45] also argue that ifXl dominatesB+
decays, thenXh dominates theB0 decays and vice-versa. Theyhave also predicted that the mass difference between theX par-ticle in B+ andB0 decays should be [45, 106]
M(Xh) − M(Xl) = (8± 3) MeV. (45)
There are two reports from Belle [39] and Babar [40] Collab-orations for the observation of theB0→ K0 X andB+ → K+ Xdecays that we show in Figs. 4 and 5.
Although the identification of theX(3872) from the decayB+ → K+ X in these two figures is very clear, this is not the casefor theB0 → K0 X decay, where the evidence for the existenceof such a state is not so clear.
In any case, if one accepts the existence of theX in the decayB0 → K0 X, these reports are not consistent with each other.While Belle measures [39]:
B(B0→ XK0)B(B+ → XK+)
= 0.82± 0.22± 0.05, (46)
and
M(X)B+ − M(X)B0 = (0.18± 0.89± 0.26) MeV, (47)
BaBar measures [40]:
B(B0→ XK0)B(B+ → XK+)
= 0.41± 0.24± 0.05, (48)
9
)2 (GeV/cXm3.8 3.82 3.84 3.86 3.88 3.9 3.92 3.94 3.96 3.98 4
)2E
ven
ts /
( 0.
005
GeV
/c
0
2
4
6
8
10
12
14
)2 (GeV/cXm3.8 3.82 3.84 3.86 3.88 3.9 3.92 3.94 3.96 3.98 4
)2E
ven
ts /
( 0.
005
GeV
/c
0
2
4
6
8
10
12
14
)2 (GeV/cXm3.8 3.82 3.84 3.86 3.88 3.9 3.92 3.94 3.96 3.98 4
)2E
ven
ts /
( 0.
005
GeV
/c
0
10
20
30
40
50
60
70
80
)2 (GeV/cXm3.8 3.82 3.84 3.86 3.88 3.9 3.92 3.94 3.96 3.98 4
)2E
ven
ts /
( 0.
005
GeV
/c
0
10
20
30
40
50
60
70
80
Figure 4: Fits to them(J/ψπ+π−) distribution for the decaysB0 → K0 X (top)andB+ → K+ X (botton) from ref.[40].
andM(X)B+ − M(X)B0 = (2.7± 1.6± 0.4) MeV. (49)
In both measurements, the mass difference between the twostates is much smaller than the prediciton in Eq.(45).
If X(3872) is a loosely bound molecular state, the branchingratio for the decayB0 → K0 X is suppresed by more than oneorder of magnitude compared to the decayB+ → K+ X. Theprediction in refs. [60, 107] gives:
0.06≤ Γ(B0→ XK0)
Γ(B+ → XK+)≤ 0.29, (50)
which is, considering the errors, still consistent with thedatafrom BaBar.
Recentely BaBar has reported the observation of the decayX(3872)→ ψ(2S)γ [26] at a rate:
B(X→ ψ(2S) γ)B(X→ ψγ)
= 3.4± 1.4, (51)
while the prediction from ref. [50] gives
Γ(X→ ψ(2S) γ)Γ(X→ ψγ)
∼ 4× 10−3. (52)
Figure 5: Fits to them(J/ψπ+π−) distribution for the decaysB0 → K0 X (top)andB+ → K+ X (botton) from ref.[39].
While this difference could be interpreted as a strong pointagainst the molecular model and as a point in favor of a conven-tional charmonium interpretation [44], it can also be interpretedas an indication that there is a significant mixing of thecc com-ponent with theD0D∗0 molecule. As a matter of fact, the neces-sity of mixing acc component with theD0D∗0 molecule was al-ready pointed out in some works [89, 108, 109, 110, 111, 112].In particular, in ref. [110] it was phenomenologically shownthat, because of the very loose binding of the molecule, the pro-duction rates of a pure moleculeX(3872) should be at least oneorder of magnitude smaller than what is seen experimentally.
In ref. [58], a Monte Carlo simulation of the productionof a boundD0D∗0 state with binding energy as small as 0.25MeV, obtained a cross section of about two orders of magni-tude smaller than the prompt production cross section for theX(3872) observed by the CDF Collaboration. The authors ofref. [58] concluded thatS-wave resonant scattering is unlikelyto allow the formation of a loosely boundD0D∗0 molecule, thuscalling for alternative (tetraquark) explanation of CDF data.However, it was pointed out in ref. [59] that a consistent anal-ysis of DD∗ molecule production requires taking into accountthe effect of final state interactions of theD and D∗ mesons.
10
This observation changes the results of the Monte Carlo calcu-lations bringing the theoretical value of the cross sectionveryclose to the observed one. Thus, the question of interpretationof X(3872) production at hadronic machines is not yet settled.
Other interpretations for theX(3872) like cusp [113], hy-brids [114, 115], or glueball [116] have already been coveredin refs. [60, 61, 62, 63, 64, 65, 66, 67]. Here we would like tofocus on the QCD sum rules studies of this meson.
3.2. QCDSR studies of X(3872)
Considering theX(3872) as aJPC = 1++ state we can con-struct a current based on diquarks in the color triplet config-uration, with symmetric spin distribution: [cq]S=1[cq]S=0 +
[cq]S=0[cq]S=1, as proposed in ref. [45]. Therefore, the corre-sponding lowest-dimension interpolating operator for describ-ing Xq as a tetraquark state is given by:
j(q,di)µ =
iǫabcǫdec√2
[(qTaCγ5cb)(qdγµCcT
e )
+ (qTaCγµcb)(qdγ5CcT
e )] , (53)
whereq denotes au or d quark.On the other hand, we can construct a current describingXq
as a molecularDD∗ state:
j(q,mol)µ (x) =
1√
2
[
(
qa(x)γ5ca(x)cb(x)γµqb(x))
−(
qa(x)γµca(x)cb(x)γ5qb(x))
]
(54)
In general, other four-quark operators with 1++ are possible.For example, starting from the simple charmed diquark statesgiven in Table II, another tetraquark current withJPC = 1++ canbe constructed by combining the pseudo scalar 0− and vector 1−
diquark. Equivalently, additional current can be constructed forthe meson type currents. The number of currents increases fur-ther, if one allows for additional color states; color sextet for thediquark and color octet for the molecular states. An extensivestudy has been carried out for the 0++ light mesons[117], withtheir mixing under renormalizations [118] from which one canform renormalization group invariant physical currents. Thechoice of the current does not matter too much provided thatwe can work with quantities less affected by radiative correc-tions and where the OPE converges quite well. This is borneout in the well-known case of baryon sum rules, where a sim-ple choice of operator [119] and a more general choice [120]have been studied. Even though apparently different, mainly inthe region of convergence of the OPE, the two choices of in-terpolating currents have provided the same predictions for theproton mass. In some cases however, particular choices mightbe preferable over the others.
3.2.1. Two-point functionFor the present case, the two currents in Eqs. (53) and (54)
were used, in refs. [121] and [122] respectively, to study the
X(3872). The correlator for these currents can be written as:
Πµν(q) = i∫
d4x eiq.x〈0|T[ j(q)µ (x) j(q)†
ν (0)]|0〉
= −Π1(q2)(gµν −qµqνq2
) + Π0(q2)qµqνq2
, (55)
where, since the axial vector current is not conserved, the twofunctions,Π1 andΠ0, appearing in Eq. (55) are independentand have respectively the quantum numbers of the spin 1 and 0mesons.
Using the current in Eq. (53), as an example, the correlationfunction in Eq. (55) can be written in terms of the quark propa-gators as:
Πµν(q) = − iεabcεa′b′c′εdecεd′e′c′
2(2π)8
∫
d4xd4p1d4p2 eix.(q−p1−p2){
Tr[
Scbb′ (p1)γ5CSqT
aa′ (x)Cγ5
]
Tr[
Sqd′d(−x)γµCScT
e′e(−p2)Cγν]
+
+ Tr[
Scbb′(p1)γνCSqT
aa′ (x)Cγµ]
×
× Tr[
Sqd′d(−x)γ5CScT
e′e(−p2)Cγ5
]
}
. (56)
In the OPE side, we work at leading order inαs and con-sider the contributions of condensates up to dimension eight.To evaluate the correlation function in Eq. (56), we use themomentum-space expression for the charm-quark propagatorgiven in Eq. (30). The light-quark part of the correlation func-tion is calculated in the coordinate-space, using the propagatorgiven in Eq. (29). The resulting light-quark part is combinedwith the charm-quark part before it is dimensionally regular-ized atD = 4.
The correlation function,Π1, in the OPE side can be writtenas a dispersion relation:
ΠOPE1 (q2) =
∫ ∞
4m2c
dsρ(s)
s− q2, (57)
where the spectral density is given by the imaginary part of thecorrelation function:πρ(s) = Im[ΠOPE
1 (s)]. For the current inEq. (53) we obtain [121]:
ρOPE(s) = ρpert(s) + ρmq(s) + ρ〈qq〉(s) + ρ〈G2〉(s)
+ ρmix(s) + ρ〈qq〉2(s) , (58)
11
with
ρpert(s) =1
210π6
αmax∫
αmin
dαα3
1−α∫
βmin
dββ3
(1− α − β)(1+ α
+β)[
(α + β)m2c − αβs
]4,
ρmq(s) = −mq
23π4
αmax∫
αmin
dαα
{
− 〈qq〉22
[m2c − α(1− α)s]2
(1− α)
+
1−α∫
βmin
dββ
[
(α + β)m2c − αβs
]
[
−m2c〈qq〉 + 〈qq〉
22
[
(α + β)m2c
− αβs]
+mc
25π2αβ2(3+ α + β)(1− α − β)
[
(α + β)m2c − αβs
]2]}
,
ρ〈qq〉(s) = −mc〈qq〉25π4
αmax∫
αmin
dαα2
1−α∫
βmin
dββ
(1+ α + β)
×[
(α + β)m2c − αβs
]2,
ρ〈G2〉(s) =
〈g2G2〉293π6
αmax∫
αmin
dα
1−α∫
βmin
dββ2
[
(α + β)m2c − αβs
]
×[
m2c(1− (α + β)2)
β− (1− 2α − 2β)
2α
[
(α + β)m2c − αβs
]
]
,
ρmix(s) =mc〈qgσ.Gq〉
26π4
αmax∫
αmin
dα[
− 2α
(m2c − α(1− α)s)
+
1−α∫
βmin
dβ[
(α + β)m2c − αβs
]
(
1α+α + β
β2
)
]
, (59)
where the integration limits are given byαmin =
(1−√
1− 4m2c/s)/2, αmax = (1+
√
1− 4m2c/s)/2 and
(βmin = αm2c)/(sα −m2
c). We have included the contribution ofthe dimension-six four-quark condensate:
ρ〈qq〉2(s) =m2
cρ〈qq〉2
12π2
√
s− 4m2c
s, (60)
and (for completeness) a part of the dimension-8 condensatecontributions:
Πmix〈qq〉1 (M2) = −
m2cρ〈qgσ.Gq〉〈qq〉
24π2
∫ 1
0dα
[
1+m2
c
α(1− α)M2
− 12(1− α)
]
exp
[
− m2c
α(1− α)M2
]
. (61)
In Eqs. (60) and (61) the parametrization in Eq. (4) was as-sumed.
Parametrizing the coupling of the axial vector meson 1++, X,to the current,j(q)
µ , in Eqs. (53) and (54) as:
〈0| j(q)µ |X〉 = λqǫµ , (62)
the phenomenological side of Eq. (55) can be written as
Πphenµν (q2) =
|λ(q)|2
M2X − q2
−gµν +qµqνM2
X
+ · · · , (63)
where the Lorentz structure projects out the 1++ state. The dotsdenote higher axial-vector resonance contributions that will beparametrized, as explained in Sec. 2.3, through the introductionof a continuum threshold parameters0. After making a Boreltransform of both sides, and transferring the continuum contri-bution to the OPE side, the sum rule for the axial vector mesonX up to dimension-eight condensates can be written as:
|λ(q)|2e−M2X/M
2=
∫ s0
4m2c
ds e−s/M2ρOPE(s) + Πmix〈qq〉
1 (M2) , (64)
1.6 1.8 2.0 2.2 2.4 2.6 2.8
1
2
3
4
5s
0
1/2 = 4.17 GeV
Pert + mq
+ <qq>
+ <g2G
2> + m
q<qq>
+ m0
2<qq>
+ <qq>2
+ m0
2<qq>
2
Condensa
te/R
HS
M2 (GeV
2)
Figure 6: Thej(q−di)µ OPE convergence in the region 1.6 ≤ M2 ≤ 2.8 GeV2 for√
s0 = 4.17 GeV (taken from ref.[121]).
For the current in Eq. (53) we show, in Fig. 6, the relativecontribution of each term on the OPE expansion of the sum rule.One can see that forM2 > 1.9 GeV2, the addition of a subse-quent term of the expansion brings the curve (representing thesum) closer to an asymptotic value (which was normalized to1). Furthermore the changes in this curve become smaller withincreasing dimension. These are the requirements for a goodOPE convergence and this fixes the lower limit of the Borelwindow to beM2 ≥ 2 GeV2.
We obtain an upper limit forM2 by imposing the constraintthat the QCD continuum contribution should be smaller thanthe pole contribution. The maximum value ofM2 for which thisconstraint is satisfied depends on the value ofs0. The compari-son between pole and continuum contributions for
√s0 = 4.15
GeV andρ = 2.1 is shown in Fig. 7.Having the Borel window fixed, the mass is obtained by us-
ing Eq. (36). In Fig. 8 we show the obtainedX mass for differentvalues of
√s0 andρ. We can see by this figure that the effect
of the violation of the factorization assumption, given byρ, issimilar to the effect of changing the continuum threshold.
To study the effect of higher dimension condensates in themass we show, in Fig. 9, the contributions of the individualcondensates toMX obtained from Eq. (36). From this figure
12
2 2.2 2.4 2.6 2.8 3M
2(GeV
2)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
pole
X c
ontin
uum
(%
)
Figure 7: The solid line shows the relative pole contribution (the pole contribu-tion divided by the total, pole plus continuum, contribution) and the dashed lineshows the relative continuum contribution for
√s0 = 4.14 GeV andρ = 2.1.
2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8M
2(GeV
2)
3.5
3.6
3.7
3.8
3.9
4
4.1
4.2
4.3
MX (
GeV
)
Figure 8: TheX meson mass as a function of the Borel parameter for differentvalues of
√s0 andρ. The solid line and the solid line with dots show the results
for√
s0 = 4.2 GeV usingρ = 1 andρ = 2.1 respectively. The dashed linesare the same for
√s0 = 4.1 GeV. The crosses indicate uper limit of the allowed
Borel window.
we see that the results oscillate around the perturbative result,and that the results obtained up to dimension-5 are very closeto the ones obtained up to dimension-8. For definiteness, thevalue ofMX obtained by including the dimension-5 mixed con-densate will be considered as the final prediction from the SR,and the effects of the higher condensates as the error due to thetruncation of the OPE.
The final result forMX, obtained in [121] considering theallowed Borel window and the uncertaities in the parameters, is
MX = (3.92± 0.13) GeV, (65)
which is compatible with the experimental value of the mass ofthe X(3872). In ref. [121] the mass difference in Eq.(45) wasalso evaluated giving:
M(Xh) − M(Xl) = (3.3± 0.7) MeV, (66)
1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8M
2(GeV
2)
3.4
3.5
3.6
3.7
3.8
3.9
4
4.1
4.2
MX (
GeV
)
Figure 9: The OPE convergence forMX in the region 1.9 ≤ M2 ≤ 2.8 GeV2 fors1/20 = 4.1 GeV. We start with the perturbative contribution plus a very smallmq
contribution (long-dashed line) and each subsequent line represents the additionof one extra condensate of higher dimension in the expansion: +〈qq〉 (solidline), +〈g2G2〉 + mq〈qq〉 (dotted-line on top the solid line),+m2
0〈qq〉 (dashedline),+〈qq〉2 (dot-dashed line),+m2
0〈qq〉2 (solid line with triangles) (taken fromref.[121]).
in agreement with the BaBar measurement in Eq. (49).In the case of the current in Eq. (54), the OPE conver-
gence and the pole contribution determine a similar Borel win-dow [122]. The result for the mass obtained in ref. [122] isMX = (3.87± 0.07) GeV, in an even better agreement with theexperimental mass. However, due to the uncertainties inherentto the QCDSR method, we can not really say that theX(3872)is better described with a molecular type of current than with atetraquark type of current. To see that we show, in Fig. 10, thedouble ratio of the sum rules
dmol/di =MXmol
MXdi
, (67)
whereMXmol andMXdi are the QCDSR results obtained by usingthe currents in Eqs. (53) and (54). The ratio plotted in this figurewas obtained by using
√s0 = 4.15 GeV and by considering the
OPE up to dimension-6.We see from Fig. 10 that the deviations between the two
QCDSR results in the allowed Borel windon are smaller than0.01%.
In ref. [123] the same currents given in Eqs. (53) and (54)were used to study the importance of including the width of thestate, in a sum rule calculation. This can be done by replac-ing the delta function in Eq. (7) by the relativistic Breit-Wingerfunction:
δ(s−m2)→ 1π
Γ√
s(s−m2)2 + sΓ2
. (68)
The mass and width are determined by looking at the stabilityof the obtained mass against varying the Borel parameterM2,as usual.
Although the effect of the width is not large, in the case ofthe molecular current, Eq. (54), it was possible to fit the exper-
13
2 2.2 2.4 2.6 2.8 3M
2(GeV
2)
0.95
0.97
0.99
1.01
1.03
1.05
d mol
/di
Figure 10: The double ratiodmol/di obtained from the QCDSR results for theXmass using the currents in Eqs. (53) and (54).
imental mass, 3872 MeV, and the widthΓ < 2.3 MeV simulta-neously with a continum threshold,
√s0 = 4.38 GeV, as can be
seen by Fig. 11.
3.8
3.85
3.9
3.95
4
2 2.2 2.4 2.6 2.8 3 3.2
mX [
GeV
]
Borel Mass M2 [GeV
2]
s0 = (4.38 GeV)2
D* D molecule
×
×
Γ=0 MeV2 MeV4 MeV6 MeV8 MeV
Exp
Figure 11: Results for theX(3872) meson mass with a molecular current. Thecrosses indicate lower and upper limit of the Borel window, respectively (takenfrom ref.[123]).
3.2.2. Three-point functionOne important question, when proposing a multiquark struc-
ture for theX(3872), is whether with a tetraquark or molecularstructure for theX(3872), it is possible to explain a total widthsmaller than 2.3 MeV. As a matter of fact, a large partial de-cay width for the decayX → J/ψπ+π− should be expected inthis case. The initial state already contains all the four quarksneeded for the decay, and there is no selection rules prohibitingthe decay. Therefore, the decay is allowed as in the case of thelight scalarsσ andκ studied in [124], where widths of the orderof 400 MeV were found. The decay widthΓ(X→ J/ψππ), withX(3872) considered as a tetraquark state, was also studied inref. [45]. The authors arrived atΓ(X → J/ψππ) ∼ 5 MeV. In
order to have such small decay width the authors had to makea bold guess about the order of magnitude of the coupling con-stant in the vertexXJ/ψV (whereV stands for theρ orω vectormeson):
gXψV = 0.475. (69)
The decay width for the decayX → J/ψV → J/ψF whereF = π+π−(π+π−π0) for V = ρ(ω) is given by [45, 125]
dΓds
(X→ J/ψF) =1
8πm2X
|M|2BV→F
× ΓVmV
π
p(s)
(s−m2V)2 + (mVΓV)2
, (70)
where
p(s) =
√
λ(m2X,m
2ψ, s)
2mX, (71)
with λ(a, b, c) = a2 + b2 + c2 − 2ab− 2ac− 2bc. The invariantamplitude squared is given by:
|M|2 = g2XψV f (mX,mψ, s), (72)
wheregXψV is the coupling constant in the vertexXJ/ψV and
f (mX,mψ, s) =13
4m2X −
m2ψ + s
2+
(m2X −m2
ψ)2
2s
+(m2
X − s)2
2m2ψ
m2X −m2
ψ + s
2m2X
. (73)
Therefore, the ratio in Eq. (40) is given by:
Γ(X→ J/ψ π+π−π0)Γ(X→ J/ψ π+π−)
=g2
XψωmωΓωBω→πππIω
g2XψρmρΓρBρ→ππIρ
, (74)
where
IV =
∫ (mX−mψ)2
(nmπ)2ds
(
f (mX,mψ, s)
× p(s)
(s−m2V)2 + (mVΓV)2
)
. (75)
Using Bω→πππ = 0.89, Γω = 8.49 GeV,mω = 782.6 MeV,Bρ→ππ = 1,Γρ = 149.4 GeV andmρ = 775.5 MeV we get
Γ(X→ J/ψ π+π−π0)Γ(X→ J/ψ π+π−)
= 0.118
(
gXψω
gXψρ
)2
. (76)
The coupling constant at theXJ/ψV vertex can be evaluateddirectly from a QCD sum rule calculation, which is based onthe three-point correlation function:
Πµνα(p, p′, q) =∫
d4xd4y eip′ .x eiq.yΠµνα(x, y), (77)
withΠµνα(x, y) = 〈0|T[ jψµ(x) jVν (y) jXα
†(0)]|0〉, (78)
14
wherep = p′ + q and the interpolating fields are given by:
jψµ = caγµca,
jρν =12
(uaγνua − daγνda), ,
jων =16
(uaγνua + daγνda). (79)
For jXα , it was shown in ref. [98] that if we use the currents inEqs.(53) or (54), the QCDSR result for the ratio between thecoupligs is given by:
gXψω
gXψρ= 1.14. (80)
Using this result in Eq. (76) we finally get
Γ(X→ J/ψ π+π−π0)Γ(X→ J/ψ π+π−)
≃ 0.15. (81)
Therefore, to be able to reproduce the experimental result inEq. (40) it is necessary to use a mixed current:
jXα = cosθ j(u)α + sinθ j(d)
α , (82)
with j(q)α given by Eq.(53) for a tetraquark current or by Eq.(54)
for a molecular current. Using the above definitions in Eq.(78),and considering the quarksu andd as degenerated, we arrive at
Πµνα(x, y) =−iNV
2√
2
(
cosθ + (−1)IV sinθ)
Πqµνα(x, y), (83)
whereNρ = Iρ = 1 andNω = 1/3, Iω = 0.To evaluate the phenomenological side of the sum rule we
insert, in Eq.(78), intermediate states forX, J/ψ andV. Usingthe definitions:
〈0| jψµ |J/ψ(p′)〉 = mψ fψǫµ(p′),
〈0| jVν |V(q)〉 = mV fVǫν(q),
〈X(p)| jXα |0〉 = λXǫ∗α(p), (84)
whereλX = (cosθ + sinθ)λq, with λq defined in Eq. (62), weobtain the following relation:
Π(phen)µνα (p, p′, q) =
i(cosθ + sinθ)λqmψ fψmV fV gXψV
(p2 −m2X)(p′2 −m2
ψ)(q2 −m2V)
×(
− ǫαµνσ(p′σ + qσ) − ǫαµσγp′σqγqν
m2V
− ǫανσγp′σqγp′µ
m2ψ
)
+ · · · ,(85)
where the dots stand for the contribution of all possible excitedstates, and the coupling constant,gXψV, is defined by the matrixelement,〈J/ψV|X〉:
〈J/ψ(p′)V(q)|X(p)〉 = gXψVǫσαµνpσǫα(p)ǫ∗µ(p
′)ǫ∗ν (q), (86)
which can be extracted from the effective Lagrangian that de-scribes the coupling between two vector mesons and one axialvector meson [45]:
L = igXψVǫµνασ(∂µXν)ΨαVσ. (87)
With the current in Eq. (82), the ratio between the couplingconstants is now given by [98]:
gXψω
gXψρ= 1.14
(
cosθ + sinθ)
(
cosθ − sinθ) . (88)
Using the previous result in Eq. (76) we get
Γ(X→ J/ψ π+π−π0)Γ(X→ J/ψ π+π−)
≃ 0.15
(
cosθ + sinθcosθ − sinθ
)2
. (89)
This is exactly the same relation obtained in refs. [45, 125],that imposesθ ∼ 200 to reproduce the experimental result inEq.(40). A similar relation was obtained in ref. [127] wherethedecay of theX into two and three pions goes through aD D∗
loop.Usingθ = 200, theXJ/ψω coupling constant was estimated
from the sum rule with a tetraquark current to be [125]:
gXψω = 13.8± 2.0, (90)
which is much bigger than the number in Eq.(69), and leads toa much bigger partial decay width:
Γ(X→ J/ψπ+π−π0) = (50± 15) MeV. (91)
A similar width was obtained in ref. [98] by using a molecularcurrent like the one in Eq. (54). Therefore, from a QCDSRcalculation it is not possible to explain the small width of theX(3872) if it is a pure four-quark state.
Considering the fact that also the relation in Eq. (51) cannot be reproduced with a pure molecular state, in ref. [98] theX(3872) was treated as a mixture between acc current and amolecular current, similar to the mixing considered in ref.[126]to study the light scalar mesons:
Jqµ(x) = sin(α) j(q,mol)
µ (x) + cos(α) j(q,2)µ (x), (92)
with j(q,mol)µ (x) given in Eq. (54) and
j(q,2)µ (x) =
1
6√
2〈qq〉[ca(x)γµγ5ca(x)]. (93)
There is no problem in reproducing the experimental mass ofthe X(3872) with this current for a large range of the mixtureangleα. Consideringα in the region 5◦ ≤ α ≤ 13◦ they get[98]:
MX = (3.77± 0.18) GeV, (94)
which is in a good agreement with the experimental value. Thevalue obtained for the mass grows with the value of the mix-ing angleα, but for α ≥ 30◦ it reaches a stable value beingcompletely determined by the molecular part of the current.
Considering also a mixture ofD+D∗− and D−D∗+ compo-nents, the most general current is given by
jXµ (x) = cosθJuµ(x) + sinθJd
µ(x), (95)
with Juµ(x) andJd
µ(x) given by Eq. (92).
15
Using the current in Eq. (95), theXJ/ψω coupling constantobtained in ref. [98] forθ = 200 and a mixing angle in Eq. (92)α = 90 ± 40 is:
gXψω = 5.4± 2.4 (96)
which gives:
Γ(
X→ J/ψπ+π−π0)
= (9.3± 6.9) MeV. (97)
The result in Eq. (97) is in agreement with the experimen-tal upper limit. It is important to notice that the width and themass grow with the mixing angleα. Therefore, there is only asmall range for the values of this angle that can provide simul-taneously good agreement with the experimental values of themass and the decay width, and this range is 5◦ ≤ α ≤ 13◦. Thismeans that theX(3872) is basically acc state with a small, butfundamental, admixture of molecularDD∗ states. By molecu-lar states we mean an admixture betweenD0D∗0, D0D∗0 andD+D∗−, D−D∗+ states, as given by Eq. (95).
In ref. [128] a similar mixture between acc state and molec-ular states (includingJ/ψρ andJ/ψω) was considered to studythe X(3872) decays intoJ/ψγ andψ(2S)γ, using effective La-grangians. In this approach the authors only needed a smalladmixture of thecc component ( equivalent toα = 780 ± 20
in Eq. (92)) to reproduce the ratio in Eq. (51). It is not clear,however, if with this smallcc admixture it is possible to obtainthe prompt production cross section for theX(3872) observedby the CDF Collaboration [58].
With the mixing anglesα andθ fixed, it is possible to evaluatethe width of the radiative decayX(3872)→ J/ψγ, to check ifit is possible to reproduce the experimental result in Eq. (39).To do that one has just to consider the current in Eq. (95) forthe X(3872), and exchange thejVν current, in Eq. (78), by thephoton currentjγν :
jγν =∑
q=u,d,c
eq qγνq , (98)
with eq =23e for quarksu andc, andeq = − 1
3e for quarkd (e isthe modulus of the electron charge).
The phenomenological side of the sum rule is given by [129]
Πphenµνα (p, p′, q) = −
(cosθ + sinθ)λqmψ fψǫµ(p′)ǫ∗α(p)
(p2 −m2X)(p′2 −mψ)
× 〈ψ(p′)| jγν |X(p)〉 , (99)
where
〈ψ(p′)| jγν(q)|X(p)〉 = i ǫγν (q)M(X(p)→ γ(q)J/ψ(p′)) , (100)
with [111]
M(X(p)→ γ(q)J/ψ(p′)) = eεκλρσǫαX(p)ǫµψ(p′)ǫργ (q) ×
× qσm2
X
(A gµλgακp · q+ Bgµλpκqα +Cgακpλqµ). (101)
In Eq. (101),A, B andC are dimensionless couplings that aredetermined by the sum rule. Using this relation in Eq.(99), the
phenomenological side of the sum rule becomes:
Πphenµνα (p, p′, q) =
ie(cosθ + sinθ)λqmψ fψm2
X(p2 −m2X)(p′2 −mψ)
×(
ǫαµνσqσA+ ǫµνλσp′λqσqαB− ǫανλσqµqσp′λC
+ ǫανλσp′λp′µqσ(C − A)p · qm2ψ
− ǫµνλσp′λqσ(qα + p′α)(A+ B)p · qm2
X
)
. (102)
The values obtained in ref. [129] for the couplings, usingθ =
20o and varyingα in the range 5o ≤ α ≤ 13o are:
A = 18.65± 0.94,
A+ B = −0.24± 0.11,
C = −0.843± 0.008. (103)
The decay width is given in terms of these couplings through:
Γ(X→ J/ψ γ) =α
3p∗5
m4X
(
(A+ B)2 +m2
X
m2ψ
(A+C)2)
,
wherep∗ = (m2X − m2
ψ)/(2mX). Using the result for the decaywidth of the channelJ/ψπ+π− in Eq. (97):Γ(X → J/ψ ππ) =9.3± 6.9 MeV, we get
Γ(X→ J/ψ γ)Γ(X→ J/ψ π+π−)
= 0.19± 0.13, (104)
which is in complete agreement with the experimental resultinEq. (39). Therefore, from a QCDSR point of view, theX(3872)is a mixture of acc state (∼97%) and molecularD0D∗0, D0D∗0
(∼2.6%) andD+D∗−, D−D∗+ (∼0.4%) states.
3.3. Summary for X(3872)
To summarize, there is an emerging consensus that theX(3872) is not a purecc state neither a pure multi-quark state.From the ratio in Eq. (40), we know thatX(3872) is not anisospin eigenstate, therefore, it can not be a purecc state. Onthe other hand, the binding energy, the production rates andtheobserved ratio in Eq. (51) are not compatible with a pure molec-ular state. Considering all the available experimental informa-tion, it is very probable that theX(3872) is a admixture of acharmonium state with other multi-quark states: molecularortetraquark states.
3.4. Predictions for Xb, Xs, Xsb
It is straightforward to extend the analysis done for theX(3872) to the case of the bottom quark. Using the same in-terpolating field of Eq. (53) with the charm quark replaced bythe bottom one, the analysis done forX(3872) was repeated forXb in ref.[121]. In this case there is also a good Borel windowand the prediction for the mass of the state that couples withatetraquark (bq)(bq) with JPC = 1++ current is:
MXb = (10.27± 0.23) GeV. (105)
16
The central value in Eq. (147) is close to the mass ofΥ(3S),and appreciably below theB∗B threshold at about 10.6 GeV.For comparison, the molecular model predicts forXb a masswhich is about 50− 60 MeV below this threshold [60], while arelativistic quark model without explicit (bb) clustering predictsa value of about 133 MeV below this threshold [130]. A futurediscovery of this state, e.g. at LHCb, will certainly test thedifferent theoretical models of this state and clarify, at the sametime, the nature of theX(3872).
In the case ofXs ([cs][ cs]) andXsb ([bs][ bs]), one has to re-
place the light quarks in the currents forX andXb by strangequarks. To extract the relatively small mass-splitting, itis ap-propriate to use the double ratio of moments [71, 131]:
dsc ≡
M2Xs
M2X
, (106)
which suppresses different systematic errors and the depen-dence on the sum rule parameters likes0 and M2. The resultobtained for this ratio in ref. [121] is:
√
dsc = 0.984± 0.007. (107)
This leads to the mass splitting:
MXs − MX ≃ −(61± 30) MeV . (108)
Similar methods used in [71, 131] have predicted successfullythe values ofMDs/MD andMBs/MB, which is not quite surpris-ing, as in the double ratios, all irrelevant sum rules systematicscancel out.
It is interesting to notice that theXs mass prediction fromref. [121] is slightly smaller than theX(3872) mass, which isquite unusual. Such a small and negative mass-splitting is ratherstriking and needs to be checked using alternative methods.The(almost) degenerate value of theX and of theXs masses maysuggest that the physically observedX(3872) state can also havea ccsscomponent.
A similar analysis can be done for theXsb ([bs][ bs]) giving
[121]:√
dSb ≡
MXsb
MXb
= 0.988± 0.018, (109)
andMXs
b− MXb = −(123± 182) MeV. (110)
We expect that theXb-family will show up at LHCb in thenear future, which will serve as a test of this predictions.
4. The Y(J PC= 1−−) family
4.1. Experiment versus theory
Thee+e− annihilation through initial state radiation (ISR) isa powerful tool to search for 1−− states at theB-factories. Thefirst state in the 1−− family discoverd using this process was theY(4260). It was first observed in 2005, by the BaBar Collabo-ration [10], in the reaction:
e+e− → γIS RJ/ψπ+π−, (111)
with massM = (4259±10) MeV and widthΓ = (88±24) MeV.TheY(4260) was confirmed by CLEO and Belle Collaborations[6].
The BaBar Collaboration also reported aππmass distributionthat peaks near 1 GeV, as can be seen in Fig. 12 [10], and thisinformation was interpreted as being consistent with thef0(980)decay.
)2) (GeV/c-π+πm(0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2
2E
vent
s / 0
.2 G
eV/c
0
20
40
60
80
Figure 12: Dipion mass distribution forY(4260)→ J/Ψπ+π− from ref. [10].
TheY(4260) was also observed in theB− → Y(4260)K− →J/Ψπ+π−K− decay [31], and CLEO reported two additional de-cay channels:J/Ψπ0π0 andJ/ΨK+K− [6].
)2)(GeV/cψJ/-π+πm(3.8 4 4.2 4.4 4.6 4.8 5 5.2 5.4
)2E
vent
s / (
20 M
eV/c
0
10
20
30
40
50
60
70
80
)2)(GeV/cψJ/-π+πm(3.8 4 4.2 4.4 4.6 4.8 5 5.2 5.4
)2E
vent
s / (
20 M
eV/c
0
10
20
30
40
50
60
70
80BABAR
preliminary
Figure 13: Theπ+π−J/Ψ invariant mass spectrum from ref. [32].
In an updated report [32], BaBar has confirmed the observa-tion of theY(4260), shown in Fig. 13. However, the newππmass distribution shows a more complex structure, as can beseem in Fig. 14.
Repeating the same kind of analysis leading to the ob-servation of theY(4260) state, in the channele+e− →γIS RΨ(2S)π+π−, BaBar [33] has identified another broad peakat a mass around 4.32 GeV, which was confirmed by Belle [12].Belle found that theψ′π+π− enhancement observed by BaBarwas, in fact, produced by two distinct peaks, as can be seen inFig. 15. The masses and widths obtained by Belle and BaBarfrom fits to Breit-Wigner resonant shapes are summarized inTable 4
17
)2)(GeV/c-π+πm(0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
)2E
vent
s / (
100
MeV
/c
0
10
20
30
40
50
60
70
80
90
100
BABARpreliminary
Figure 14: Dipion mass distribution forY(4260)→ J/Ψπ+π− from ref. [32].
Table 4: Masses and widths ofY(1−−) states, measured by Belle and BaBar
Bellestate Decay mode M ( MeV) Γ( MeV)
Y(4260) J/ψπ+π− 4247± 12+17−32 108± 19± 10
Y(4360) ψ(2S)π+π− 4361± 9± 9 74± 15± 10Y(4660) ψ(2S)π+π− 4664± 11± 5 48± 15± 3
BaBarstate Decay mode M ( MeV) Γ( MeV)
Y(4260) J/ψπ+π− 4259± 6+2−3 105± 18+4
−6Y(4360) ψ(2S)π+π− 4324± 24 172± 33
As it is evident from Fig. 15, there is no sign of theY(4260) intheψ′π+π− mass spectrum. Theππ mass distribution reportedin [12] for these two resonances can be seen in Fig. 16.
All these three states share the following properties: theyarevector states with the photon quantum numbers, they have largetotal widths, and they have only been observed in charmoniumdecay modes.
Since the masses of these states are higher than theD(∗)D(∗)
threshold, if they were 1−− charmonium states they should de-cay mainly toD(∗)D(∗). However, the observedY states do notmatch the peaks ine+e− → D(∗)±D(∗)∓ cross sections measuredby Belle [34] and BaBar [35, 36]. Besides, theΨ(3S), Ψ(2D)andΨ(4S) cc states have been assigned to the well stablishedΨ(4040), Ψ(4160), andΨ(4415) mesons respectively. Thepredictions from quark models for theΨ(3D) andΨ(5S) char-monium states are 4.52 GeV and 4.76 GeV respectively. There-fore, the masses and widths of these three newY states arenot consistent with any of the 1−− cc states [63, 64, 132], andtheir discovery represents an overpopulation of the charmoniumstates, as it can be seen in Fig. 17.
An interesting interpretation is that theY(4260) is a charmo-nium hybrid. Hybrids are hadrons in which the gluonic degreeof freedom has been excited. The nature of this gluonic excita-tion is not well understood, and has been described by variousmodels. The spectrum of charmonium hybrids has been calcu-lated using lattice gauge theory [133]. The result for the mass isapproximately 4200 MeV, which is consistent with the flux tube
Figure 15: Evidence for the vector statesY(4360) andY(4660) from ref. [12].
0
2
4
6
0.4 0.6 0.8M(π+π-) (GeV/c2)
Ent
ries/
20 M
eV/c
2(a)
02468
1012
0.4 0.6 0.8 1M(π+π-) (GeV/c2)
Ent
ries/
20 M
eV/c
2
(b)
Figure 16: Dipion mass distribution for the vector statesY(4360) (a) andY(4660) (b) from ref. [12].
model predictions [134]. However, more recent lattice simula-tions [135] and QCD string models calculations [136], predictthat the lightest charmonium hybrid has a mass of 4400 MeV,which is closer to the mass of theY(4360). In any case, a predic-tion of the hybrid hypothesis is that the dominant open charmdecay mode would be a meson pair with oneS-waveD meson(D, D∗, Ds, D∗s) and oneP-waveD meson (D1, Ds1) [137]. Inthe case of theY(4260) this suggests dominance of the decaymodeDD1. Therefore, a largeDD1 signal could be understoodas a strong evidence in favor of the hybrid interpretation for theY(4260). In the case of theY(4360) andY(4660), since theirmasses are well above theDD1 threshold, their decay intoDD1
should be very strong if they were charmonium hybrids.A critical information for understanding the structure of these
states is wether the pion pair comes from a resonance state.From the di-pion invariant mass spectra shown in Figs. 14 and16, there is some indication that only theY(4660) has a welldefined intermediate state consistent withf0(980) [138]. Dueto this fact and the proximity of the mass of theψ′ − f0(980)system with the mass of theY(4660) state, in ref. [139], theY(4660) was considered as af0(980) ψ′ bound state. If thisinterpretation of theY(4660) is correct, heavy quark spin sym-metry implies that there should be aη′c − f0(980) bound state[140]. This state would decay mainly intoη′cππ, and the authorsof ref. [140] predicted the mass of such state to be 4616+5
−6 MeV.
18
Figure 17: Masses of the 1−− cc states from quark models and the masses oftheY states.
The observation of this new state would clearly determine thestructure of theY(4660). TheY(4660) was also suggested to bea baryonium state [141], a canonical 53S1 cc state [142], and atetraquark with asc-scalar- diquark and a ¯sc-scalar-antidiquarkin a 2P-wave state [143].
In the case ofY(4260), in ref. [46] it was considered as asc-scalar-diquark ¯sc-scalar-antidiquark in aP-wave state. Ma-iani et al. [46] tried different ways to determine the orbitalterm and they arrived atM = (4330± 70) MeV, which is moreconsistent withY(4360). However, from theππ mass distri-bution in refs. [32, 12], none of these two states,Y(4260) andY(4360) has a decay with an intermediate state consistent withf0(980) and, therefore, it is not clear that they should have ansspair in their structure. Besides, in ref. [143] the authors showthat the mass of a [sc]S=0[ sc]S=0 tetraquark in aP-wave statewould be 200 MeV higher than theY(4260) mass. The authorsof ref. [143], found that a more natural interpretation for theY(4260) would be a [qc]S=0[qc]S=0 tetraquark in aP-wave state.TheY(4260) was also interpreted as a baryoniumΛc − Λc state[144], a S-wave threshold effect [145], as a resonance due tothe interaction between the three,J/ψππ andJ/ψKK, mesons[146] and as a 4S charmonium state [147]. Although there aresome arguments against the molecular interpretation [60, 148],theY(4260) was also considered as a molecular state bound bymeson exchange [149, 150, 151]. The threeY states were alsointerpreted as non-resonant manifestations of the Regee zeros[152].
4.2. QCDSR studies for the Y(JPC = 1−−) states
The Y(JPC = 1−−) states can be described by molecular ortetraquark currents, with or without asspair. In refs. [153, 154]a QCD sum rule calculation was performed using these kind ofcurrents.
The lowest-dimension interpolating operator to describea JPC = 1−− state with the symmetric spin distribution:[cs]S=0[cs]S=1 + [cs]S=1[cs]S=0 is given by:
jµ =ǫabcǫdec√
2[(sT
aCγ5cb)(sdγµγ5CcTe )
+ (sTaCγ5γµcb)(sdγ5CcT
e )] . (112)
2,8 3,2 3,6 4,0 4,4
0,0
2,0x10-7
4,0x10-7
6,0x10-7
8,0x10-7
OPE
(GeV
10)
M2 (GeV2)
Pert. <ss> <g2G2> <sg Gs> <ss>2
<ss> <sg Gs> Total
Figure 18: Thejµ OPE convergence in the region 2.8 ≤ M2 ≤ 4.5 GeV2 for√s0 = 5.1 GeV (taken from ref.[153]).
From Fig. 18 we see that the OPE convergence forM2 ≥ 3.2GeV2 is better than the one shown in Fig. 6, since the perturba-tive contribution is the dominant one in the whole Borel region.Using the dominance of the pole contribution to fix the uppervalue of the Borel window, the result for the mass of the statedescribed by the current in Eq. (112) obtained in ref. [153] is:
mY = (4.65± 0.10) GeV, (113)
in excellent agreement with the mass of theY(4660) me-son. Therefore, the conclusion in ref. [153] is that the mesonY(4660) can be described with a diquark-antidiquark tetraquarkcurrent with a spin configuration given by scalar and vectordiquarks. The quark content in the current in Eq. (112) isalso consistent with the di-pion invariant mass spectra shownin Fig. 16, which shows that there is some indication that theY(4660) has a well defined di-pion intermediate state consis-tent with f0(980). It is also very interesting to notice that thef0(980) meson may be itself considered as a tetraquark state[155]. This aspect should play an important role in theY(4660)decay since, as shown in ref. [125], it is very hard to explainasmall decay width when the intital four-quark state decays intotwo final two-quark states.
Replacing the strange quarks in Eq. (112) by a generic lightquarkq the mass obtained for a 1−− state described with thesymmetric spin distribution: [cq]S=0[cq]S=1 + [cq]S=1[cq]S=0 is[153]:
mY = (4.49± 0.11) GeV, (114)
which is bigger than theY(4350) mass, but is consistent with itconsidering the uncertainty.
19
TheY mesons can also be described by molecular-type cur-rents. In particular aDs0(2317)D∗s(2110) molecule withJPC =
1−−, could also decay intoψ′π+π− with a dipion mass spectraconsistent withf0(980). A current withJPC = 1−− and a sym-metrical combination of scalar and vector mesons is given by:
jµ =1√
2[( saγµca)(cbsb) + (caγµsa)(sbcb)] . (115)
The mass obtained in ref. [153] for the current in Eq. (115) is
mDs0D∗s = (4.42± 0.10) GeV, (116)
which is more in agreement with theY(4350) mass than withtheY(4660) mass.
To consider a molecularD0D∗ current withJPC = 1−−, onehas only to change the strange quarks in Eq.(115) by a genericlight quarkq. The mass obtained with such current is [153]
mD0D∗ = (4.27± 0.10) GeV, (117)
in excellent agreement with the mass of the mesonY(4260).Again, in order to conclude if we can associate this molecularstate with the mesonY(4260) we need a better understandingof the di-pion invariant mass spectra for the pions in the de-cayY(4260)→ J/ψπ+π−. From the spectra given in Fig. 14, itseems that theY(4260) is consistent with a non-strange molecu-lar stateD0D∗. Using aD0 mass [21]mD0 = 2352±50 MeV, theD0D∗ threshold is around 4360 MeV and it is 100 MeV abovethe mass in Eq. (117), indicating the possibility of a bound state.
A JPC = 1−− molecular current can also be constructed withpseudoscalar and axial-vector mesons. A molecularDD1 cur-rent was used in ref. [154]. The mass obtained with this currentis:
mDD1= (4.19± 0.22) GeV. (118)
Therefore, considering the errors, the molecularDD1 assigne-ment for the mesonY(4260) is also possible, in agreement withthe findings of ref. [149], where a meson exchange model wasused to study theY(4260) meson. theDD1 threshold is around4285 MeV and very close to theY(4260) mass, indicating thepossibility of a loosely bound molecular state.
4.3. Summary for Y(JPC = 1−−) states
To summarize, the discovery of theY(4260), Y(4360) andY(4660) appears to represent an overpopulation of the expectedcharmonium 1−− states. The absence of open charm produc-tion is also inconsistent with a conventionalcc explanation.Possible explanations for these states include charmoniumhy-brid, tetraquark state andD0D∗ or DD1 molecular state forY(4260). TheY(4360) could be a charmonium hybrid, or atetraquark state (with two axial [cs] diquarks inP-wave or withtwo scalar [cs] diquarks inP-wave). TheY(4660) could be acanonical 53S1 cc state, or a tetraquark state (with symmetrical[cs]S=1[cs]S=0 or with [cs]S=0[cs]S=0 in a 2P-wave).
From the QCDSR results described is this Section, theY(4260) could be aD0D∗ or a DD1 molecular state and theY(4660) could be a tetraquark state with symmetrical spin dis-tribution: [cs]S=1[cs]S=0 + [cs]S=0[cs]S=1. It is not possible
to describe theY(4360) neither as aDs0D∗s molecular state,nor as a tetraquark state with symmetrical spin distribution:[cq]S=1[cq]S=0 + [cq]S=0[cq]S=1.
5. The Z+(4430) meson
All states discussed so far are electrically neutral. The realturning point in the discussion about the structure of the newobserved charmonium states was the observation by Belle Col-laboration of a charged state decaying intoψ′π+, produced inB+ → Kψ′π+ [13].
5.1. Experiment versus theory
The measured mass and width of this state areM = (4433±4± 2) MeV andΓ = (45+18+30
−13−13) MeV [13]. TheB meson decayrate to this state is similar to that for the decays to theX(3872)andY(3930) mesons. There are no reports of aZ+ signal in theJ/ψπ+ decay channel. Since the minimal quark content of thisstate isccud, this state is a prime candidate for a multiquark me-son. TheZ+(4430) was observed in theψ′π+ channel, therefore,it is an isovector state with positiveG-parity: IG = 1+.
Using the same data sample as in ref. [13], Belle also per-formed a full Dalitz plot analysis [37] and has confirmed the ob-servation of theZ+(4430) signal with a 6.4σ peak significance,as can be seen in Fig. 19. The updatedZ+(4430) parametersare:M = (4433+15+19
−12−13) MeV andΓ = (109+86+74−43−56) MeV.
M2(π+ψ’), GeV2/c4
Eve
nts
/ 0.1
8 G
eV2 /c
4
0
10
20
30
40
50
60
14 16 18 20 22
Figure 19: Dalitz plot projection for theψ′π+ invariant mass from ref. [37]. Thesolid (dotted) histograms shows the fit result with (without) a singleψ′π+ state.The dashed histogram represents the background.
Babar Collaboration [38] also searched theZ−(4430) signa-ture in four decay modes:B → ψπ−K, whereψ = J/ψ or ψ′
andK = K0S or K+. No significant evidence for a signal peak
was found in any of the processes investigated, as it can be seenin Fig. 20.
The Babar result gives no conclusive evidence of theZ+(4430) seen by Belle.
There are many theoretical interpretations of theZ+(4430)structure [156, 157, 158, 159, 160, 161, 162, 47, 163, 164, 165,166, 167, 168, 169, 170, 171, 172]. Because its mass is close totheD∗D1 threshold, Rosner [156] suggested that it is anS-wavethreshold effect, while others considered it to be a strong candi-date for aD∗D1 molecular state [157, 158, 159, 160, 161, 162].
20
0
500
1000 (a) 0,+K-πψ J/→-,0B
Events-πAll K
0
200
400(d) 0,+K
-π(2S)ψ →-,0B Events-πAll K
0
500
(b)(1430)
2
*(892) + K
*K
100
200
300
400 (e)(1430)
2
*(892) + K
*K
) 2 (GeV/c-πψJ/m3.5 4 4.5
0
100
200
(c) veto
*K
) 2 (GeV/c-π(2S)ψm3.8 4 4.2 4.4 4.6 4.8
0
100
200 (f) veto
*K
2E
vent
s/10
MeV
/c
Figure 20: TheJ/ψπ− (left) andψ′π− (right) invariant mass from ref. [38].
Other possible interpretations are tetraquark state [47, 163], acusp in theD∗D1 channel [164], a baryonium state [165], aradially excitedcs state [166], or a hadro-charmonium state[66]. The tetraquark hypothesis implies that theZ+(4430)will have neutral partners decaying intoψ′π0/η. Besidesthe spectroscopy, there are discussions about its production[156, 168, 169, 170] and decay [171].
Considering theZ+(4430) as a loosely boundS-waveD∗D1
molecular state, the allowed angular momentum and parity areJP = 0−, 1−, 2−, although the 2− assignment is probably sup-pressed in theB+ → Z+K decay by the small phase space.Among the remaining possible 0− and 1− states, the formerwill be more stable as the later can also decay toDD1 in S-wave. Moreover, one expects a bigger mass for theJP = 1−
state as compared to aJP = 0− state. There is also a quenchedlattice QCD calculation that finds attractive interatiction in theD∗D1 system in theJP = 0− channel [173]. The authors ofref. [173] also find positive scattering lenght. Based on thesefindings, they conclude that although the interaction betweenthe two charmed mesons is attractive in this channel, it is un-likely that they can form a genuine bound state right below thethreshold.
5.2. QCDSR calculations for Z+(4430)
ConsideringZ+(4430) as aD∗D1 molecule withJP = 0−, apossible current describing such state, considered in ref.[158],is given by:
j =1√
2
[
(daγµca)(cbγµγ5ub) + (daγµγ5ca)(cbγ
µub)]
. (119)
This current corresponds to a symmetrical stateD∗+D01+D∗0D+1 ,
and has positiveG-parity, which is consistent with the observeddecayZ+(4430)→ ψ′π+.
The mass obtained in a QCDSR calculation using such a cur-rent was [158]:
mD∗D1 = (4.40± 0.10) GeV, (120)
in an excellent agreement with the experimental mass.
To check if theZ+(4430) could also be described as adiquark-antidiquark state withJP = 0− , the following currentwas considered in ref. [162]:
j0− =iǫabcǫdec√
2[(uT
aCγ5cb)(ddCcTe ) − (uT
aCcb)(ddγ5CcTe )] .
(121)The mass obtained with this current was [162]
mZ(0−)= (4.52± 0.09) GeV, (122)
which is a little bigger than the experimental value [13], butstill consistent with it, considering the uncertainties. Compar-ing this result with the result in Eq. (120), we see that the re-sult obtained using a molecular-type current is in a better agree-ment with the experimental value. As mentioned in Sec. II, thisstrongly suggests that the state is better explained as a molec-ular state than as a diquark-antidiquark state. This is explicitlyborne out in the calculation for the coupling,λ, between thestate and the current defined in Eq. (8) of Sec. II. We get:
λZ(0−)= (3.75± 0.48) × 10−2 GeV5, (123)
λD∗D1 = (5.66± 1.26) × 10−2 GeV5. (124)
Therefore, one can conclude that the physical particle withJP = 0− and quark contentccud couples with a larger strengthwith the molecularD∗D1 type current than with the current inEq. (121).
In ref. [162] it was also considered as a diquark-antidiquarkinterpolating operator withJP = 1− and positiveG parity:
j1−
µ =ǫabcǫdec√
2[(uT
aCγ5cb)(ddγµγ5CcTe )
+ (uTaCγ5γµcb)(ddγ5CcT
e )] . (125)
In this case the Borel stability obtained is worse than for theZ+
with JP = 0− [162], and the obtained mass was
mZ(1−)= (4.84± 0.14) GeV, (126)
which is much bigger than the experimental value and biggerthan the result obtained using the current withJP = 0− inEq. (122). From these results it is possible to conclude that,while it is also possible to describe theZ+(4430) as a diquark-antidiquark state or a molecular state withJP = 0−, theJP = 1−
configuration is disfavored.
5.3. Summary for Z+(4430)
A confirmation of the existence of theZ±(4430) is criticalbefore a complete picture can be drawn. If confirmed, theonly open options for theZ+(4430) structure are tetraquark ormolecule. The favored quantum numbers areJP = 0−.
5.4. Sum rule predictions for B∗B1 and D∗sD1 molecules
It is straightforward to extend the analysis done for theD∗D1
molecule to the case of the bottom quark. Using the same inter-polating field of Eq. (119) with the charm quark replaced by thebottom one, the analysis done forZ+(4430) was repeated forZb
21
in ref.[158]. The OPE convergence in this case is even betterthan the one forZ+(4430) and the predicted mass is
mZB∗B1= (10.74± 0.12) GeV, (127)
in a very good agreement with the prediction in ref. [174].In the case of the strange analogous mesonZ+s considered as
a pseudoscalarD∗sD1 molecule, the current is obtained by ex-changing thed quark in Eq. (119) by thesquark. The predictedmass is [158]:
mD∗sD1 = (4.70± 0.06) GeV, (128)
which is bigger than theD∗sD1 threshold∼ 4.5 GeV, indicatingthat this state is probably a very broad one and, therefore, itmight be very dificult to be seen experimentally.
6. The Z+1
(4050) and Z+2
(4250) states
The Z+(4430) observation motivated studies of otherB0 →K−π+(cc) decays. In particular, the Belle Collaboration has re-ported the observation of two resonance-like structures intheπ+χc1 mass distribution [7].
6.1. Experiment versus theoryThe two resonance-like structures, calledZ+1 (4050) and
Z+2 (4250), were observed in the exclusive processB0 →K−π+χc1. The significance of each of theπ+χc1 structures ex-ceeds 5σ and, if they are interpreted as meson states, their min-imal quark content must beccud. Since they were observed inthe π+χc1 channel, the only quantum numbers that are knownabout them areIG = 1−.
M (χc1π+), GeV/c2
Eve
nts
/ 0.0
24 G
eV/c
2
0
5
10
15
20
25
30
35
40
3.6 3.8 4 4.2 4.4 4.6 4.8
Figure 21: Theπ+χc1 invariant mass distribution from ref. [7]. The solid(dashed) histograms shows the fit result with two (without any) π+χc1 reso-nance. The dots with error represent data.
When fitted with two Breit-Wigner reconance amplitudes,the resonance parameters are:
M1 = (4051± 14+20−41) MeV,
Γ1 = (82+21+47−17−22) MeV,
M2 = (4248+44+180−29− 35) MeV,
Γ2 = (177+54+316−39− 61) MeV. (129)
The invariant mass distribution, where the contribution ofthestructure in theπ+χc1 channel is most clearly seen, is shown inFig. 21
Before this observation, it was suggested, in ref. [51], thatresonances decaying intoχc1 and one or two pions are expectedin the framework of the hadro-charmonium model.
Due to the closeness of theZ+1 (4050) andZ+2 (4250) masses tothe D∗D∗(4020) andD1D(4285) thresholds, these states couldalso be interpreted as molecular states or threshold effects. Liuet al. [175], using a meson exchange model find strong attrac-tion for theD∗D∗ system withJP = 0+, while using a bosonexchange model, the author of ref. [176] concluded that the in-terpretation ofZ+1 (4050) as aD∗D∗ molecule is not favored.In any case, it is very difficult to understand a bound molecu-lar state which mass is above theD∗D∗ threshold. In the caseof Z+2 (4250), using a meson exchange model, it was shown inref. [149] that its interpretation as aD1D or D0D∗ molecule isdisfavored.
6.2. QCDSR calculations
In ref. [154], the QCD sum rules formalism was used tostudy theD∗D∗ andD1D molecular states withIGJP = 1−0+
and 1−1− respectively. The currents used in both cases are:
jD∗D∗ = (daγµca)(cbγµub) , (130)
and
jµ =i√
2
[
(daγµγ5ca)(cbγ5ub) + (daγ5ca)(cbγµγ5ub)]
. (131)
The mass obtained with the current in Eq. (130) is [154]:
mD∗D∗ = (4.15± 0.12) GeV (132)
where the central value is around 130 MeV above theD∗D∗(4020) threshold, indicating the existence of repulsive in-teractions between the twoD∗ mesons. Strong interaction ef-fects might lead to a repulsion that could result in a virtualstateabove the threshold. Therefore, this structure may or may notindicate a resonance.
For the current in Eq. (131), the mass obtained is [154]:
mD1D = (4.19± 0.22) GeV, (133)
where the central value is around 100 MeV below theD1D(4285) threshold, and, considering the errors, consistentwith the mass of theZ+2 (4250) resonance. Therefore, in thiscase, there is an attractive interaction between the mesonsD1
andD which can lead to a molecular state.In ref. [123] it was found that the inclusion of the width, in
the phenomenological side of the sum rule, increases the ob-tained mass for molecular states. This means that the intro-duction of the width in the sum rule calculation, increases themass of the states that couple to theD∗D∗ and D1D molec-ular currents. As a result, using the current in Eq. (131), itis possible to obtain a massmD1D = 4.25 GeV with a width40≤ Γ ≤ 60 MeV, as can be seen in Fig. 22.
22
3.8
4
4.2
4.4
4.6
4.8
mD
1D
[G
eV
]
D1D molecule
s0 = (4.3 GeV)2
10%20%
70%60%50%Γ=0 MeV
20 MeV40 MeV60 MeV80 MeV
100 MeVExp
s0 = (4.4 GeV)2
10%20%
70%60%50%
3.6
3.8
4
4.2
4.4
4.6
4.8
2.2 2.4 2.6 2.8 3 3.2
mD
1D
[G
eV
]
Borel Mass M2 [GeV
2]
s0 = (4.5 GeV)2
10%20%
60%50%
2.2 2.4 2.6 2.8 3 3.2
Borel Mass M2 [GeV
2]
s0 = (4.6 GeV)2
10%20%
60%50%
Figure 22: Results for theD1D molecule from ref. [123]. Each panel shows adifferent choice of the continuum threshold. Upward and downward arrows in-dicate the region of the Borel windowM2
min andM2max, respectively. Associated
numbers in % denote the dimension eight condensate contribution for upwardarrows and continuum contribution for downward ones.
On the other hand, the mass of theD∗D∗ molecule will be farfrom theZ+1 (4050) mass. Therefore, the authors of ref. [154]conclude that it is possible to describe theZ+2 (4250) resonancestructure as aD1D molecular state withIGJP = 1−1− quantumnumbers, and that theD∗D∗ state is probably a virtual state thatis not related with theZ+1 (4050) resonance-like structure. Con-sidering the fact that theD∗D∗ threshold (4020) is so close tothe Z+1 (4050) mass and that theη′′c (31S0) mass is predicted tobe around 4050 MeV [65], it is probable that theZ+1 (4050) isonly a threshold effect [65].
7. The Y(3930) and Y(4140) states
7.1. Experiment versus theory
The Y(3930) was first observed by the Belle Collaboration[4] in the decayB→ KY(3930)→ KωJ/ψ. It was confirmedby BaBar [41, 83] in two channelsB+ → K+ωJ/ψ andB0 →K0ωJ/ψ. The measured mass from these two Collaborationsare: (3943± 11) MeV from ref. [4] and (3919.1+3.8
−3.4 ± 2) MeVfrom ref. [83], which gives an average mass of (3929±7) MeV.This state has positiveC andG parities and the total width is(31+10−8 ±5) MeV [83]. ThemJ/ψω mass distribution observed by
BaBar is shown in Fig. 23, from where we also see theX(3872)observed in its decay intoJ/ψω.
Since the decayY → J/ψω is OZI suppressed for a char-monium state [67], also forY(3930) it was conjectured that itcould be a hybridccg state [137], a molecular state [50, 109,175, 178, 179, 177], or a tetraquark state [45].
A recent acquisition to the list of peculiar states is the nar-row structure observed by the CDF Collaboration in the decayB+ → Y(4140)K+ → J/ψφK+. The particle’s signature peakcan be seem in Fig. 24. The mass and width of this structure isM = (4143± 2.9 ± 1.2) MeV, Γ = (11.7+8.3
−5.0 ± 3.7) MeV [8].Since theY(4140) decays into twoIG(JPC) = 0−(1−−) vectormesons, like theY(3930) it has positiveC andG parities. ThepossibleJP quantum numbers of aS-wave vector-vector sys-tem are 0+, 1+, 2+. However, sinceC = (−1)L+S, theJP = 1+
is forbidden for a state withL = 0 andC = +1. Therefore,
4 4.2 4.4 4.6 4.8
2E
vent
s/10
MeV
/c
0
200
400
a)
3.85 3.9 3.950
200
400 c)
)2 (GeV/cωψJ/m4 4.2 4.4 4.6 4.8
2E
vent
s/10
MeV
/c
0
200
400
b) DataFit functionX(3872)Y(3940)Nonresonant
Figure 23: Reconstruction of theX(3872) andY(3930) peaks from their decaysinto J/ψω mesons by BaBar (taken from ref. [83]).
the possible quantum numbers forY(3930) andY(4140) areJPC = 0++, 1−+ and 2++. At these quantum numbers, 1−+ isnot consistent with the constituent quark model and it is con-sidered exotic.
Figure 24: Reconstruction of theY(4140) peak from its decay into muons andK mesons by CDF.
There are already some theoretical interpretations of thisstructure. Its interpretation as a conventionalcc state is disfa-vored because, as pointed out by the CDF Collaboration [8], itlies well above the threshold for open charm decays and, there-fore, acc state with this mass would decay predominantly intoan open charm pair with a large total width. It was also shownin ref. [180] that theY(4140) is probably not aP-wave charmo-nium state:χ
′′
cJ (J = 0, 1). If it were the case, the branchingratio of the hidden charm decay,Y(4140)→ J/ψφ, would be
23
much smaller than the experimental observation.In ref. [181], the authors interpreted theY(4140) as the
molecular partner of the charmonium-like stateY(3930). Theyconcluded that theY(4140) is probably aD∗sD
∗s molecular
state withJPC = 0++ or 2++, while theY(3930) is itsD∗D∗
molecular partner. This idea is supported by the fact that themass difference between these two mesons is approximatelythe same as the mass difference between theφ andω mesons:mY(4140) − mY(3930) ∼ mφ − mω ∼ 210 MeV. It is also inter-esting to notice that, if theY(4140) and theY(3930) mesonsareD∗sD
∗s and D∗D∗ molecular states, the binding energies of
these states will be approximately the same:mY(4140)− 2mD∗s ∼mY(3930)−2mD∗ ∼ −90 MeV. However, with a meson exchangemechanism to bind the two charmed mesons, it seems naturalto expect a more deeply bound system in the case that pionscan be exchanged between the two charmed mesons, as in theD∗D∗, than when onlyη andφ mesons can be exchanged, as intheD∗sD
∗s system.
In ref. [182] the author argue that theY(4140) can be inter-preted either as aD∗sD
∗s molecular state or as an exotic hybrid
charmonium withJPC = 1−+. A molecularD∗sD∗s configuration
was also considered in refs. [183, 184, 185, 186, 187]. UsingQCD sum rules, the authors of refs. [185, 187] agree that it ispossible to describe theY(4140) with a molecularD∗sD
∗s current
with JPC = 0++. Using one boson exchange model the author ofref. [186] showed that the effective potencial of theD∗sD
∗s sys-
tem supports the explanation ofY(4140) as a molecular state.In ref. [188] theY(4140) was considered as aJPC = 1++ ccsstetraquark state. TheJPC = 1++ assignment reduces the cou-pling of theY(4140) with the vector-vector channel and, there-fore, a small decay width would be possible in this case but notfor a JPC = 0++ tetraquark state, as pointed out in ref. [181].The authors of ref. [189] argue that theJ/ψφ system has quan-tum numbersJPC = 1−− and that the enhancement observed byCDF does not represent any kind of resonance. There is alsoa prediction for the radiative open charm decay of theY(4140)that could test the molecular assignment of this state [190].
7.2. QCDSR calculation for Y(3930)and Y(4140)
Considering theY(3930) andY(4140) asIGJPC = 0+0++
states, the possible currents that couple with aD∗D∗ andD∗sD∗s
molecular states are
jq = (qaγµca)(cbγµqb) , and js = (saγµca)(cbγ
µsb) , (134)
respectively. These two currents were considered in a QCDSRstudy in ref. [185]. Surprisingly the masses obtained were
mD∗sD∗s = (4.14± 0.09) GeV, (135)
andmD∗D∗ = (4.13± 0.11) GeV. (136)
There is another QCDSR calculation for theD∗D∗ molecularcurrent withJPC = 0++ [191], that finds a massmD∗D∗ = (3.91±0.11) GeV, compatible with theY(3930) state. However, inref. [191] the authors have considered only the condensatesupto dimension six. We show, in Fig. 25, the contribution of all
Figure 25: The OPE convergence for theD∗D∗ molecular current, as a functionof the Borel mass, for
√s0 = 4.4 GeV. We plot the relative contributions starting
with the perturbative one and each other line represents thetotal OPE afteradding of one extra condensate in the expansion.
the terms in the OPE side of the sum rule, up to dimension-six. From this figure we see that only forM2 ≥ 3.5 GeV2
the contribution of the dimension-six condensate is less than20% of the total contribution. Therefore, the lower value ofM2 in the sum rule window should beM2
min = 3.5 GeV2. Theinclusion of the dimension-eight condensate improves the OPEconvergence, and its contribution is less than 20% of the totalcontribution forM2 ≥ 2.5 GeV2, as it can be seen in Fig. 26.
2.2 2.4 2.6 2.8 3 3.2M
2(GeV
2)
0.5
0.7
0.9
1.1
1.3
1.5
1.7
OP
E
Figure 26: The OPE convergence for theD∗D∗ current for√
s0 = 4.6 GeV. Weplot the relative contributions starting with the perturbative one (long-dashedline), and each other line represents the total OPE after adding of one extra con-densate in the expansion:+ 〈ss〉 (dashed line),+ 〈g2G2〉 (dotted line),+m2
0〈ss〉(dot-dashed line),+ 〈ss〉2 (line with circles),+ m2
0〈ss〉2 (line with squares).From ref. [185].
Besides, we observe that considering only condensates up todimension-six, there is no pole dominance, in the Borel rangeconsidered in ref. [191], as it can be seen in Fig. 27 and, there-fore, no allowed Borel window can be found in this case.
In the case of theD∗sD∗s molecular current, the inclusion of
the dimension-eight condensate almost does not change theOPE convergence and the value of the mass. This is the rea-
24
3,0 3,5 4,0 4,5 5,00,0
0,2
0,4
0,6
0,8
1,0
pole
x c
ontin
num
(%)
M2 (GeV2)
Pole Continuum
Figure 27: The pole and the continuum contributions for theD∗D∗ current for√s0 = 4.4 GeV.
son why the results obtained with theD∗sD∗s molecular current
are the same in refs. [185] and [187].Therefore, from a QCD sum rule study, the results obtained
in refs. [185] and [187] indicate that theY(4140) narrow struc-ture observed by the CDF Collaboration in the decayB+ →Y(4140)K+ → J/ψφK+ can be very well described by a scalarD∗sD
∗s current. The mass obtained with theD∗D∗ scalar current,
on the other hand, depends on the dimension of the condensatesconsidered in the OPE side, showing that there is still no OPEconvergence in the sum rule up to dimension-6 condensate. Totest if the convergence is achieved up to dimension-8, it is im-portant to consider higher dimension condensates in the OPEside of the sum rule.
8. The X(3915) and X(4350) states
In recent communications, the Belle Collaboration has re-ported the observation of two narrow peaks in the two-photonprocessesγγ → ωJ/ψ [2] andγγ → φJ/ψ [11], as can be seenin Figs. 28 and 29.
W (GeV)
Events
/10 M
eV
Figure 28: TheωJ/ψ mass enhancement observed by Belle Coll. in theγγ →ωJ/ψ events. From ref. [192].
These two states were reported in the experimental reviews[68, 192, 193, 194]. When fitted with Breit-Wigner resonance
0
2
4
6
8
4.2 4.4 4.6 4.8 5
M(φJ/ψ) (GeV/c2)
Ent
ries/
25 M
eV/c
2
Figure 29: TheφJ/ψ mass enhancement observed by Belle Coll. in theγγ →φJ/ψ events. From ref. [192].
amplitudes, the resonance parameters are:
M = (3914± 4± 2) MeV,
Γ = (28± 12+2−8) MeV, (137)
for X(3915), and
M = (4350.6+4.6−5.1± 0.7) MeV,
Γ = (13.3+17.9−9.1 ± 4.1) MeV. (138)
for X(4350).Since these two states decay into two vector mesons, they
have positiveC andG parities, like theY(3930) andY(4140).As a matter of fact, it is possible that theX(3915) be the samestate as theY(3930), observed by Belle [4] and BaBar [41, 83]in the decay channelB → KY(3930) → KωJ/ψ, since theBabar mass and width for this state are very close to the result inEq. (137):M = (3919.1+3.8
−3.4±2 MeV andΓ = (31+10−8 ±5) MeV.
In any case, a charmonium assignment for this state is difficult[68].
In the case of theX(4350) its mass is much higher than theY(4140) mass and, as it can be seen in Fig. 29, noY(4140) sig-nal is observed in the two-photon processγγ → φJ/ψ. Thisfact was interpreted, in ref. [192], as a point against theD∗+s D∗−smolecular picture for theY(4140) states. As shown in ref. [184],aD∗+s D∗−s molecular state should be seen in the two-photon pro-cess.
The possible quantum numbers for a state decaying intoJ/ψφareJPC = 0++, 1−+ and 2++. At these quantum numbers, 1−+
is not consistent with the constituent quark model and it is con-sidered exotic. In ref. [11] it was noted that the mass of theX(4350) is consistent with the prediction for acscs tetraquarkstate withJPC = 2++ [188] and aD∗+s D∗−s0 molecular state [195].However, the state considered in ref. [195] hasJP = 1− withno definite charge conjugation. A molecular state with a vectorand a scalarDs mesons with negative charge conjugation wasstudied by the first time in ref. [153], and the obtained mass was(4.42± 0.10) GeV, also consistent with theX(4350) mass, butwith not consistent quantum numbers. A molecular state withavector and a scalarDs mesons with positive charge conjugationcan be constructed using the combinationD∗+s D∗−s0 − D∗−s D∗+s0 .
25
Some possible interpretations for this state are: a excitedP-wave charmonium stateΞ′′c2 [196]; a mixed charmonium-D∗sD
∗s
state [197].
8.1. QCDSR calculation for X(4350)
Following Belle Collaboration’s suggestion [11], inref. [198], a QCDSR calculation using aD∗sD
∗s0 current with
JPC = 1−+, was considered to test if the new observed reso-nance structure,X(4350), can be interpreted as such molecularstate.
A current that couples with aJPC = 1−+ D∗sD∗s0 molecular
state is given by:
jµ =1√
2
[
(saγµca)(cbsb) − (caγµsa)(sbcb)]
. (139)
For this current the OPE convergence is very good and thedimension-8 condensate (obtained using the factorizationhy-pothesis) is almost negligible, as can be seen by Fig. 30.
2.8 3.3 3.8 4.3 4.8M
2(GeV
2)
0.7
1.2
1.7
2.2
OP
E
Figure 30: The OPE convergence for theJPC = 1−+, D∗sD∗s0 molecule inthe region 2.8 ≤ M2 ≤ 4.8 GeV2 for
√s0 = 5.3 GeV. We plot the relative
contributions starting with the perturbative contribution plus dems correction(long-dashed line), and each other line represents the relative contribution af-ter adding of one extra condensate in the expansion:+ 〈ss〉 + ms〈ss〉 (dashedline), + 〈g2G2〉 (dotted line),+ 〈sgσ.Gs〉 + ms〈sgσ.Gs〉 (dot-dashed line),+〈ss〉2+ms〈ss〉2 (line with circles),+ 〈ss〉〈sgσ.Gs〉+ms〈ss〉〈sgσ.Gs〉 (line withsquares). Taken from ref. [198].
Considering condensates up to dimension-8 and keepingterms which are linear in the strange quark massms, the massobtained in ref. [198] was:
mD∗sD∗s0= (5.05± 0.19) GeV, (140)
where the uncertainty was obtained by considering the QCDparameters in the ranges:mc = (1.23 ± 0.05) GeV, ms =
(0.13 ± 0.03) GeV, 〈qq〉 = −(0.23 ± 0.03)3 GeV3, m20 =
(0.8 ± 0.1) GeV2, and also by allowing a violation of the fac-torization hypothesis by usingρ = 2.1.
The result in Eq. (140) is much bigger than the mass of thenarrow structureX(4350) observed by Belle. Therefore, the
authors of ref. [198] concluded that it is not possible to inter-pret theX(4350) as aD∗sD
∗s0 molecular state withJPC = 1−+.
It is also interesting to notice that the mass obtained for astate described with a 1−−, D∗sD
∗s0 molecular current, given in
Eq. (116), is much smaller than the result obtained with the1−+, D∗sD
∗s0 molecular current. This may be interpreted as an
indication that it is easier to form molecular states with not ex-otic quantum numbers.
9. The X(3940), Z(3930), X(4160) and Y(4008) states
The X(3940) [5] and theX(4160) [9] were observed by theBelle Collaboration in the analysis of theMrecoil(J/ψ) recoilspectrum ine+e− → J/ψ(cc). TheX(3940) mass and width are(3942+7
−6±6) MeV andΓ = (37+26−15±8) MeV, and was observed to
decay intoD∗D. TheX(4160) was found to decay dominantlyinto D∗D∗ with parameters given asM = (4156+25
−20± 15) MeVandΓ = (139+111
−61 ± 21) MeV. The only other known charmo-nium sate observed ine+e− → J/ψX process hasJ = 0. Thisfact and the absence of these states in theDD decay suggeststhat these states favorJPC = 0−+. However, these states are ei-ther too low or too high to be theη′′c or theη′′′c states [199]. TheX(4160) was identified as a 2++ state in two recent works: gen-erated from a relativistic four quark equations in ref.[200], andfrom a coupled channel approach using a vector-vector interac-tion in ref. [177]. In ref. [201] two assignments were found tobe possible:ηc(4S) and the P-wave excited stateχc0(3P). Sofar, no other serious theoretical attempts were performed to in-vestigate these states, and the nature of these states remains apuzzle.
TheZ(3930) was observed by Belle collaboration as an en-hancement in theγγ → DD event. The observed mass andwidth areM = 3929± 5 ± 2 MeV andΓ = 29± 10± 2 MeV.The measured properties are consistent with expectations forthe previously unseenχ′c2 charmonium state[202, 84].
The Y(4008) was seen by the Belle collaboration as a by-product while confirming theY(4260) in a two resonance fit tothee+e− → π+π−J/ψ reaction [6]. There are theoretical specu-lations that this state might be theψ(3S) or is aD∗D∗ molecularstate [203, 176]. However, BaBar could not so far confirm ex-perimentally its existence [32].
10. Other multiquark states
10.1. A DsD∗ molecular state
If the mesonsX(3872),Z+(4430),Y(4260) andZ+2 (4250) arereally molecular states, then many other molecules should ex-ist. A systematic study of these molecular states and their ex-perimental observation would confirm its structure and providea new testing ground for QCD within multiquark configura-tions. In this context, a natural extension would be to probethe strangeness sector. In particular, in analogy with the mesonX(3872), aDsD∗ molecule withJP = 1+ could be formed in theB meson decayB → πXs → π(J/ψKπ). Since it would decayinto J/ψK∗ → J/ψKπ, it could be easily reconstructed.
26
In ref. [122] the QCD sum rules approach was used to pre-dict the mass of theDsD∗ molecular state. Such prediction is ofparticular importance for new upcoming experiments which caninvestigate with much higher precision the charmonium energyregime, like the PANDA experiment at the antiproton-protonfacility at FAIR, or a possible Super-B factory experiment.Es-pecially PANDA can do a careful scan of the various thresholdsbeing present, in addition to going through the exact form ofthe resonance curve.
The current used in ref. [122] is very similar to the current inEq. (54) for theX(3872):
jµ =1√
2
[
(saγ5ca)(cbγµdb) − (saγµca)(cbγ5db)]
. (141)
The mass obtained for this state is [122]:
MDsD∗ = (3.96± 0.10) GeV, (142)
which is around 100 MeV bigger than the mass of theX(3872)meson and below theDsD∗ threshold (M(DsD∗) ≃ 3980 MeV).
1.8 2 2.2 2.4 2.6 2.8 3 3.2M
2(GeV
2)
3.6
3.7
3.8
3.9
4
4.1
4.2
4.3
4.4
mD
sD* (
GeV
)
X
X
X
Figure 31: TheDsD∗ meson mass as a function of the sum rule parameter (M2)for√
s0 = 4.4 GeV (dashed line),√
s0 = 4.5 GeV (solid line) and√
s0 = 4.6GeV (dot-dashed line). The crosses indicate the upper limitin the Borel regionallowed by the dominance of the QCD pole contribution (takenfrom ref.[122]).
The Borel curve for the mass of such state is quite stable, ascan be seen in Fig. 31, and has a minimum within the relevantBorel window. Such a stable Borel curve strongly suggests thepossibility of the existence of aDsD∗ molecular state withJP =
1+.Of course if such state exists, there will be no reason why
its isospin partners would not exist as well. Therefore, chargedstates with the quark contentccsu andccus, and with a massgiven in Eq. (142), should also be observed in the decay chan-nels:J/ψK∗+ andJ/ψK∗−.
10.2. A[cc][ ud] state
Considering that the double-charmonium production was al-ready observed in the reaction [5]
e+e− → J/ψ + X(3940), (143)
it seems that it would be possible to form the tetraquark[cc][ ud]. Such state with quantum numbersI = 0, J = 1 andP = +1 which, following ref.[204], we callTcc, is especially in-teresting. Although the process in Eq. (143) will create twoccpairs, the production ofTcc will further involve combining twolight anti-quarks withccand might be suppressed. On the otherhand, heavy ion collisions at LHC might provide another op-portunity for its non-trivial production[205]. As alreadynotedpreviously [204, 206], theTcc state cannot decay strongly orelectromagnetically into twoD mesons in theS wave due toangular momentum conservation nor inP wave due to parityconservation. If its mass is below theDD∗ threshold, this decayis also forbidden, and this state would be very narrow.
ConsideringTcc as an axial diquark-antidiquark state, a pos-sible current describing it is given by:
jµ =i2ǫabcǫdec[(cT
aCγµcb)(udγ5CdTe )
= i[cTaCγµcb][ uaγ5CdT
b ] . (144)
This current represents well the most attractive configurationexpected with two heavy quarks. This is so because the mostattractive light antidiquark is expected to be the in the colortriplet, flavor anti-symmetric and spin 0 channel [207, 208,209]. This is also expected quite naturally from the color mag-netic interaction, which can be phenomenologically parameter-ized as,
Vi j = −CH
mimjλi · λ jσi · σ j . (145)
Here, m, λ, σ are the mass, color and spin of the constituentquarki, j. Depending on the color state, the color factorλa
i λaj is
− 83 ,
43 ,−
163 for quark quark in the color state3, 6 and for quark-
antiquark in the singlet state respectively. The phenomenolog-ical value ofCH can be estimated from fits to the baryon andmeson mass splitting, from which one finds that two constantsCM andCB can fit all the mass splitting within the mesons andthe baryons respectively. Also, one finds thatCM is about afactor of 3/2 larger thanCB[205, 210]. Therefore, favorablemultiquark configuration will inevitably involve diquark con-figuration with spin zero in color3: this is a scalar diquarkwith anti symmetric flavor combination due to anti-symmetryof the total wave function involving color, spin and flavor. Thetetraquark picture for the light scalar nonet involves states com-posed of two scalar diquarks. However, here, one notes thatthe two pseudo scalar mesons will have a smaller mass and thetetraquark state not stable against strong decay. Possiblestableconfigurations emerge if either the diquark or the anti-diquarkin the tetraquark is composed of heavy quarks; this is so becausethe large attraction between quark-antiquark will be suppressedby the heavy quark mass while the strong attraction in eitherthe diquark or the anti-diquark will remain. Simple estimateswithin a constituent quark model predict stable configurationswith spin zero, where a light diquark combines with a scalar(cb) diquark, and with spin one, where a light diquark com-bines with spin 1 heavy diquark [210]. TheTcc falls into thelatter case where heavy anti-diquark is a color tripletcc with
27
spin 1, as the pair should combine antisymmetrically. Althoughthe spin 1 configuration is repulsive, its strength is much smallerthan that of the light diquark due to the heavy charm quarkmass. Therefore a constituent quark picture forTcc would bea light anti-diquark in color triplet, flavor anti-symmetric andspin 0 (ǫdec[udγ5CdT
e ]) combined with a heavy diquark of spin1 (ǫabc[cT
aCγµcb]). The simplest choice for the current whichhas a non zero overlap with such aTcc configuration is givenin Eq. (144). While a similar configurationTss is also possible[211], we believe that the repulsion in the strange diquark withspin 1 will be larger and hence energetically less favorable.
There are some predictions for the masses of theTQQ states.In ref. [212] the authors use a color-magnetic interaction,with flavor symmetry breaking corrections, to study heavytetraquarks. They assume that the Belle resonance,X(3872),is acqcq tetraquark, and use its mass as input to determine themass of other tetraquark states. They getMTcc = 3966 MeVand MTbb = 10372 MeV. In ref. [204], the authors use one-gluon exchange potentials and two different spatial configu-rations to study the mesonsTcc and Tbb. They getMTcc =
3876− 3905 MeV andMTbb = 10519− 10651 MeV. Thereare also calculations using expansion in the harmonic oscilla-tor basis [213], and variational method [214]. In ref. [210], theauthors use Eq. (145) together with a simple diquark pictureto find stable heavy tetraquark configuration with spin zeroTcb
and spin oneTcc,Tcb,Tbb.In ref. [215] a QCD sum rule calculation was done with the
current in Eq. (144). The authors found that the results are notvery sensitive to the value of the charm quark mass, neither tothe value of the condensates. The QCDSR predictions for theTcc mesons mass is:
MTcc = (4.0± 0.2) GeV, (146)
in a very good agreement with the predictions based on theone gluon exchange potential model [204], and color-magneticmodel [212].
It is straightforward to extend the analysis done for theTcc
to the the bottom sector. The prediction for theTbb mass fromref. [215] is:
MTbb = (10.2± 0.3) GeV , (147)
also in a very good agreement with the results in refs. [204],[212] and [214].
The results from ref. [215] show that while theTcc mass isbigger than theD∗D threshold at about 3.875 GeV, theTbb
mass is appreciably below theB∗B threshold at about 10.6 GeV.Therefore, these results indicate that theTbb meson shouldbe stable with respect to strong interactions and must decayweakly. These results also confirm the naive expectation thatthe exotic states with heavy quarks tend to be more stable thanthe corresponding light states [216].
11. Summary
We have presented a review, from the perspective of QCDsum rules, of the new charmonium states recently observed byBaBar and Belle Collaborations. As it was seen case by case,
this method has contributed a great deal to the understanding ofthe structure of these new states. When a state is first observedand its existence still needs confirmation, a QCDSR calculationcan be very useful. It can provide evidence in favor or againstthe existence of the state. We have computed the masses ofseveralX,Y andZ states and they were supported by QCDSR. Insome cases a tetraquark configuration was favored and in someother cases a molecular configuration was favored. QCDSRdoes not always corroborate previous indications. In Ref. [217],a comprehensive QCDSR analysis of light tetraquarks led tothe conclusion that their existence (as tetraquarks states) is veryunlikely. However, a more positive conclusion was found in thecase of tetraquarks with at least one heavy quark.
The limitations in statements made with QCDSR estimatescome from uncertainties in the method. However these state-ments can be made progressively more precise as we knowmore experimental information about the state in question.Onegood example is theX(3872), from which, besides the mass,several decay modes were measured. Combining all the avail-able information and using QCDSR to calculate the observeddecay widths, we were able to say that theX(3872) is a mixedstate, where the most important component is acc pair, whichis mixed with a small molecular component, out of which alarge fraction is composed by neutralD and D∗ mesons withonly a tiny fraction of chargedD and D∗ mesons. This con-clusion is very specific and precise and it is more elaboratedthan the other results presented addressing only the massesofthe new charmonia. This improvement was a consequence ofstudying simultaneously the mass and the decay width. Thistype of combined calculation will eventually be extended toallstates.
We close this review with some conclusions from the resultspresented in the previous sections. They are contained in Tables5 and 6.
In Table 5 we present a summary of the most plausible inter-pretations for some of the states presented in Table 1, describedin the previous sections.
Table 5: Structure and quantum numbers from QCDSR studies.
state structure JPC
X(3872) mixedcc− DD∗ 1++
Y(4140) D∗sD∗s molecule 0++
Z+2 (4250) DD1 molecule 1−
Y(4260) D0D∗ or DD1 molecule 1−−
Z+(4430) D∗D1 molecule 0−
Y(4660) [cs][ cs] tetraquark 1−−
In Table 6 we present some predictions for bottomoniumstates, that may be observed at LHCb.
From the last Table we see that the mass predictions for thestatesXb andTbb are basically the same. These states, if theyexist, may be produced at LHCb. Since theTbb is a chargedstate, it could be more easily identified. Moreover, since itsmass prediction lies below theBB∗ threshold, it should be verynarrow because it could not decay strongly.
28
Table 6: Prediction for bottomonium states from QCDSR studies.
state structure JPC mass (GeV)Xb [bq][ bq] tetraquark or 1++ 10.27± 0.23
BB∗ moleculeZb [bq][ bq] tetraquark or 0− 10.74± 0.12
B∗B1 moleculeTbb [bb][ ud] tetraquark 1++ 10.2± 0.3
BB∗ molecule
Looking at Table 6 we see that the predictions made withtetraquark and molecular currents are the same. Probably, inorder to distinguish one from the other, we will need to knowthe decay width of these states. On the other hand, from thepoint of view of measurement, this convergence of predictionsmakes them robust. In other words, even not knowing very pre-cisely their structure, we know that these states must be there.
Table 5 represents the final result of a comprehensive effortand a careful analysis of several theoretical possibilities in thelight of existing data. It is an encouraging example of whatQCDSR can do. This Table contains a short summary of whatwe have learned about the new charmonium states in the recentpast. Table 6 addresses the near future and shows some predic-tions.
From what was said above, hadron spectroscopy is and willbe a very lively field and QCDSR is a powerful tool to study it.
Acknowledgements: The authors would like to thank S.Narison, J.-M. Richard, U. Wiedner, R.M. Albuquerque, M.E.Bracco, R.D. Matheus, K. Morita, R. Rodrigues da Silva andC.M. Zanetti, with whom they have collaborated in one or moreof the works described in this review. The authors are indebtedto the brazilian funding agencies FAPESP and CNPq, and to theKorea Research Foundation KRF-2006-C00011.
12. Appendix: Fierz Transformation
In general, the currents constructed from diquark antidiquarktype of fields are related to those composed of meson type offields. However, the relation are suppressed by a typical colorand Dirac factors so that we obtain a reliable sum rule onlyif we have chosen the current well to have a maximum overlapwith the physical meson. This is expected to be particularlytruefor multiquark configuration with special diquark or molecularstructures. Therefore, if we obtain a sum rule that reproducesthe physical mass well, we can infer the inner structure of themultiquark configuration. For example, if the state is of molec-ular type, it will have a maximum overlap with a current that isconstructed with the two corresponding meson current; the sumrule will then be able to reproduce the physical mass well. Onthe other hand, if a current with a diquark-antidiquark typeofcurrent is used, the overlap to the physical meson will be small
and the sum rule will not be able to reproduce the mass well.The opposite will also be true.
To show the suppression, we discuss Fierz transformation oftetraquark current made of diquark fields into quark-antiquarkfields. The starting relation is given as follows.
qiaq
jb = −
18λαab
[
δi j (qλαq) + γµi j (qγµλαq)
−(γµγ5)i j (qγµγ5λαq) + γ5i j (qγ
5λαq)
+12σµν
i j (qσµνλαq)]
− 112δab
[
δi j (qq) + γµi j (qγµq)
−(γµγ5)i j (qγµγ5q) + γ5i j (qγ
5q)
+12σµν
i j (qσµνq)]
. (148)
Here, a, b are the color andi, j the Dirac spinor indices re-spectively. The two color representations come from 3
⊗
3 =8⊕
1.
In the formulas to follows, the two color terms can be workedout using the following formulas,
ǫabcǫdec(qTaΓ1λ
αbdc
Te )(qλαΓ2c) = −(cλαΓT
1 q)(qλαΓ2c)
ǫabcǫdec(qTaΓ1δbdc
Te )(qλαΓ2c) = 2(cΓT
1 q)(qΓ2c) (149)
We start by applying the formula to the current composed oftwo scalar diquarks. After Fierz transformation, we find,
jdi = ǫabcǫdec(qTaCγ5cb)(qdγ5CcT
e )
= (−18
)[
− (cλαq)(qλαc) − (cλαγµq)(qλαγµc)
−(cλαγµγ5q)(qλαγµγ5c)
−(cλαγ5q)(qλαγ5c) +12
(cλασµνq)(qλασµνc)]
+(16
)[
− (cq)(qc) − (cγµq)(qγµc)
−(cγµγ5q)(qγµγ5c)
−(cγ5q)(qγ5c) +12
(cσµνq)(qσµνc)]
. (150)
This is a typical example of the expansion. The factor of 1/6multiplying the color singlet quark-antiquark types of currents,comes from the color and Dirac factors and is responsible forthe small overlap to the various quark-antiquark type of cur-rents. In Eq. (150), all meson type of currents contributes.In general however, depending on the current type and chargeconjugation, only certain types of meson-meson currents con-tributes. Several examples for the expansion used in the text aregiven below.
29
1. ForX(3872) withJPC = 1++, we find,
j(q,di)µ =
iǫabcǫdec√2
[(qTaCγ5cb)(qdγµCcT
e )
+(qTaCγµcb)(qdγ5CcT
e )]
=i√
2(−1
8)(−1)2
[
∑
Γ
(
cλα(γµCΓTCγ5
+γ5CΓTCγµ)q
)
(qλαΓc)]
+i√
2(− 1
12)(2)(−1)
[
∑
Γ
(
c(γµCΓTCγ5
+γ5CΓTCγµ)q
)
(qΓc)]
j(q,di)µ =
i√
2(−1
8)[
2(cλαγ5q)(qλαγµc)
−2(cλαγµq)(qλαγ5c) − 2i(cλασµνq)
×(qλαγνγ5c) + 2i(cλαγνγ5q)(qλασµνc)]
+i√
2(16
)[
2(cγ5q)(qγµc)
−2(cγµq)(qγ5c) − 2i(cσµνq)(qγνγ5c)
+2i(cγνγ5q)(qσµνc)]
. (151)
As can be seen from the above equation, the molecu-
lar component as given inj(q,mol)µ = i√
2
[
(cγ5q)(qγµc) −
(cγµq)(qγ5c)]
comprises only a small part of the wave
function given in Eq. (151). The overlap to the molecu-lar type is suppressed by13 so that the overall contributionto the correlation function is suppressed by1
9.2. ForZ+(4430)
• with JP = 0−, we find the following current workswell.
j(di)0− =
iǫabcǫdec√2
[(uTaCγ5cb)(ddCcT
e )
−(uTaCcb)(ddγ5CcT
e )]
=i√
2(−1
8)[
2(cλαγµγ5u)(dλαγµc)
+2(cλαγµu)(dλαγµγ5c)]
+i√
2(16
)[
2(cγµγ5u)(dγµc)
+2(cγµu)(dγµγ5c)]
. (152)
As can be seen from the above equation, the molecu-lar component as given in Eq. (119), contributes witha suppression factor of 1/3. It is useful to comparethis factor to the ratio of the overlaps found in QCDsum rule analysis and given in Eqs. (123) and (124);
numerically, the ratioλZ0−/λD∗D1 = 0.66. On theother hand, assuming that theZ+(4430) has only amolecular component that couples dominantly to themolecular current, the ratio should be 1/3, as can beseen from Eq. (152). The fact that it is larger than1/3 suggest that while theZ+(4430) is dominantly ofmolecular type, there still remains some coupling toother type of non-molecular type of current compo-nent, such as the diquark-antidiquark type.
• with JP = 1−, we find the following current does notfit any of the observedZ states.
jµ =iǫabcǫdec√
2[(uT
aCγ5cb)(ddγµγ5CcTe )
+(uTaCγµγ5cb)(ddγ5CcT
e )]
=i√
2(−1
8)[
− 2(cλαγ5u)(dλαγµγ5c)
−2(cλαγµγ5u)(dλαγ5c)
−2i(cλασµνu)(dλαγνc) − 2i(cλαγνu)(dλασµνc)]
+i√
2(16
)[
− 2(cγ5u)(dγµγ5c) − 2(cγµγ5u)(dγ5c)
−2i(cσµνu)(dγνc) − 2i(cγνu)(dσµνc)]
. (153)
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